#6775092

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	#30398	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1500 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)(3,6)(4,5)(7,8)(17,18,21,23)(19,26,28,32)(20,25,22,29)(24,30,31,27);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(18,23)(24,31)(25,29)(26,32);(10,13)(14,16)(17,19)(18,24)(20,27)(21,28)(22,30)(23,31)(25,32)(26,29);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,18)(19,24)(20,29)(21,23)(22,25)(26,27)(28,31)(30,32);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,21,22)(18,25,23,29)(19,27,28,30)(24,32,31,26)

Polycyclic presentation:

2^9|a^2=i,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,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, 6682288);