#6809853

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1669 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,27,25)(23,28,26,34)(24,31,33,36)(29,35,32,30);(2,3)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,27,26)(22,28,25,34)(24,30,33,35)(29,36,32,31);(1,2,4,3)(21,24,27,33)(22,29,25,32)(23,30,26,35)(28,36,34,31);(6,11)(10,13)(12,19)(18,20)(21,25,27,22)(23,28,26,34)(24,32,33,29)(30,36,35,31);(1,3,4,2)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(22,25)(28,34)(29,32)(31,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=hi,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,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, 6717049);