#224492

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=eh,d^2=e,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=de,g^c=gh

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 131688);