#6745601

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2281 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6652797);