#72306303

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

5633 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|b^2=g,c^2=fghi,e^2=j,f^2=j,g^2=i,h^2=ij,b^a=bf,c^a=ch,c^b=cj,d^a=dj,d^c=di,e^b=egh,e^c=efgi,f^a=fj,f^b=fi,f^c=fi,g^a=gij,g^e=gij,h^a=hij,h^b=hi,h^c=hi,h^e=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, 3, F.6*F.7*F.8*F.9);;
gap> SetPower(coll, 5, 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, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.7*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, 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, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);