#72341018

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32899
commutator, Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
prime radical	#8751831	C₂
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2315 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

3236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=f,c^2=h,d^2=i,g^2=i,b^a=be,c^b=cefhi,d^a=dhj,d^b=dg,d^c=dhj,e^d=eh,f^c=fj,f^d=fi,g^a=gh,g^c=ghi,g^d=gi,g^e=gj,h^b=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341048
maximal normal subgroup of	#72341048

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