#6464641

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1479 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)(17,18,22,26,23,27,21,25)(19,28,30,20,29,31,32,24);(9,10)(11,15)(12,13)(14,16)(17,19,23,29)(18,24,27,20)(21,30,22,32)(25,28,26,31);(2,5)(6,8)(10,13)(14,15)(19,29)(20,24)(28,31)(30,32);(9,11,12,16)(10,14,13,15)(17,20,23,24)(18,19,27,29)(21,28,22,31)(25,32,26,30);(1,3,4,7)(2,6,5,8)(17,21,23,22)(18,25,27,26)(19,30,29,32)(20,31,24,28)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,d^c=di,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, 6371837);