#6757317

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	#35564	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1813 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,27,22,28,21,32)(19,25,20,30,29,24,31,26);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19)(18,24)(20,23)(21,31)(22,29)(25,28)(26,27)(30,32);(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,23)(21,29)(22,31)(24,28)(26,27)(30,32);(2,6)(5,7)(10,13)(14,16)(17,20)(18,26)(19,21)(22,31)(23,29)(24,32)(25,27)(28,30);(3,8)(5,7)(17,21,22,23)(18,27,28,32)(19,20,29,31)(24,30,25,26)

Polycyclic presentation:

2^9|a^2=g,c^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,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, 6664513);