#460507

order	512
nilpotency class	3
isoclinism	isoclinic class of #454045
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1424 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ei,c^2=e,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,h^c=hi,f^d=fi

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, 367703);