#6757616

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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1504 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)(17,18,21,24,22,23,30,31)(19,20,26,27,25,28,32,29);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,25)(18,20,23,28)(21,26,30,32)(24,27,31,29);(9,11,12,16)(10,13,14,15)(17,20)(18,19)(21,29)(22,28)(23,25)(24,32)(26,31)(27,30);(2,6)(5,7)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,31)(30,32);(3,8)(5,7)(10,14)(13,15)(18,23)(19,25)(24,31)(26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=hi,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, 6664812);