#72306295

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁵
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

5546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(2,6)(3,10)(4,13)(7,15)(11,12)(14,16)(17,18)(19,28)(20,25)(21,35)(22,26)(23,27)(24,32)(29,31)(30,45)(33,41)(34,42)(36,38)(37,40)(39,46)(43,44)(47,48);(1,2,4,8,5,6,13,9)(3,7,12,16,11,14,10,15)(17,19,22,32,23,29,21,36)(18,24,26,31,27,38,35,28)(20,30,33,47,34,44,37,46)(25,39,41,45,42,48,40,43);(3,11)(7,14)(10,12)(15,16)(19,29)(24,38)(28,31)(30,44)(32,36)(39,48)(43,45)(46,47);(1,3,5,11)(2,7,6,14)(4,12,13,10)(8,16,9,15)(17,20)(18,25,27,42)(19,30)(21,37)(22,33)(23,34)(24,39,38,48)(26,41,35,40)(28,43,31,45)(29,44)(32,47)(36,46);(1,3,4,10,5,11,13,12)(2,7,9,16,6,14,8,15)(17,18)(19,31,29,28)(20,25,34,42)(21,35)(22,26)(23,27)(24,32,38,36)(30,43)(33,41,37,40)(39,47)(44,45)(46,48)

Polycyclic presentation:

2^10|b^2=g,d^2=fghj,e^2=hj,f^2=j,g^2=i,h^2=ij,b^a=bf,c^b=cj,d^a=dh,d^c=dij,e^a=ehj,e^b=eghi,e^c=eij,e^d=efg,f^a=fj,f^b=fi,f^d=fi,f^e=fi,g^a=gij,g^e=gij,h^a=hij,h^b=hi,h^d=hi

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306341
maximal normal subgroup of	#72306341

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 4, F.6*F.7*F.8*F.10);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);