#468087

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

1476 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)(17,18)(19,23)(20,24)(21,22)(25,32)(26,29)(27,31)(28,30);(2,5)(6,8)(9,10,12,14)(11,15,16,13)(18,22)(23,29)(24,30)(31,32);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,27)(21,26)(22,29)(24,32)(25,28)(30,31);(3,7)(6,8)(17,20)(18,24)(19,25)(21,28)(22,30)(23,31)(26,27)(29,32)

Polycyclic presentation:

2^9|a^2=ei,b^2=g,c^2=g,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, 375283);