#6865510

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#39136	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1620 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6772706);