#6762168

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#39335	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1837 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)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,30,43)(20,24,32,39)(21,35,28,25)(23,36,38,31)(29,42,33,46)(34,48,44,41)(37,47,40,45);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(20,32)(24,39)(29,33)(34,44)(37,40)(41,48)(42,46)(45,47);(10,14)(11,15)(17,19,28,36,22,30,21,31)(18,23,35,43,26,38,25,27)(20,33,44,45,32,29,34,47)(24,40,48,42,39,37,41,46);(2,6)(5,7)(17,20)(18,24)(19,29)(21,34)(22,32)(23,37)(25,41)(26,39)(27,42)(28,44)(30,33)(31,45)(35,48)(36,47)(38,40)(43,46);(3,8)(5,7)(11,15)(13,16)(19,30)(20,32)(23,38)(24,39)(27,43)(31,36)(34,44)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6669364);