#6388830

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#432 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1672 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,21,25,22,20,29,24)(19,23,28,31,27,26,32,30);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,10)(14,15)(17,19,21,28,22,27,29,32)(18,23,25,31,20,26,24,30);(17,20)(18,22)(19,26)(21,24)(23,27)(25,29)(28,30)(31,32);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(19,27)(23,26)(28,32)(30,31);(1,5,6,4)(2,8,9,14)(3,11,12,13)(7,16,10,15)(19,27)(23,26)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=dfi,e^b=ei,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296026);