#458015

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

1264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,20)(17,19)(21,22,25,29)(23,31,34,27)(24,28,26,30)(32,36,33,35);(6,9)(10,12)(13,15,17,16)(14,18,19,20)(21,23)(22,27)(24,33)(25,34)(26,32)(28,36)(29,31)(30,35);(5,7,8,11)(6,10,9,12)(13,16,17,15)(14,18,19,20)(23,32)(24,26)(27,35)(28,30)(31,36)(33,34);(2,4)(7,11)(10,12)(21,24,25,26)(22,28,29,30)(23,33,34,32)(27,36,31,35)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=efi,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=de,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, 365211);