#6469261

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1593 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)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18,24,28,23,19,22,27)(20,25,30,32,21,26,29,31);(9,10)(11,16)(12,14)(13,15)(17,19,23,18)(20,25,21,26)(22,27,24,28)(29,32,30,31);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,20)(18,25)(19,26)(21,23)(22,29)(24,30)(27,31)(28,32);(10,14)(13,16)(17,21,23,20)(18,26,19,25)(22,30,24,29)(27,32,28,31);(1,4,5,3)(2,6,7,8)(20,21)(25,26)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=g,d^2=i,e^2=ghi,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376457);