#6463692

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1763 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)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,23,27,22,19,21,25)(20,24,30,32,28,26,29,31);(2,5)(6,7)(10,14)(11,12)(17,19,22,18)(20,26,28,24)(21,25,23,27)(29,31,30,32);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(9,12,13,11)(10,15,14,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(20,28)(24,26)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,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, 6370888);