#6660893

order	512
nilpotency class	3
isoclinism	isoclinic class of #6660408
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	#23920	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1862 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,21,26)(19,28,30,23)(20,32,31,25)(22,24,29,27);(3,7)(5,8)(9,10,12,13)(11,14,15,16)(17,19,21,30)(18,23,26,28)(20,22,31,29)(24,32,27,25);(1,3)(2,5)(4,7)(6,8)(10,13)(14,16)(18,24)(20,31)(22,29)(23,32)(25,28)(26,27);(3,7)(5,8)(9,11,12,15)(10,14,13,16)(17,20,21,31)(18,25,26,32)(19,29,30,22)(23,27,28,24);(2,6)(5,8)(18,26)(19,30)(20,31)(24,27)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=di,e^b=ei,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, 6568089);