#72285602

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|b^2=f,c^2=h,e^2=i,b^a=bh,c^a=ch,c^b=cg,d^a=dfgi,d^c=dfj,e^a=ehj,e^b=ef,e^c=ej,e^d=ei,f^a=fj,f^c=fi,g^a=gj,g^b=gi,g^c=gj,g^e=gi,h^b=hj,h^d=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286185, #72310874, #72310940, …
maximal normal subgroup of	#72286185, #72310874, #72310940, …

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.6);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);