#406482

order	512
nilpotency class	3
isoclinism	isoclinic class of #394267
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1320 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=h,c^2=f,d^2=e,b^a=be,c^a=cf,d^a=dh,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, 313678);