#6660739

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660418
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	#24382	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1819 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6567935);