#406618

order	512
nilpotency class	3
isoclinism	isoclinic class of #394329
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1061 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=fi,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313814);