#466819

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(2,3)(5,6,8,9)(7,10,11,12)(13,14)(15,20)(16,30)(17,35)(18,23)(19,24)(21,32)(22,31)(25,26)(27,43)(28,38)(29,39)(33,42)(34,37)(36,41)(40,44);(6,9)(10,12)(13,15,26,44,18,28,19,29)(14,20,25,40,23,38,24,39)(16,31,34,27,32,36,33,17)(21,41,42,35,30,22,37,43);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,27)(17,28)(18,32)(19,34)(20,35)(22,39)(23,30)(24,42)(25,37)(26,33)(29,36)(31,44)(38,43)(40,41);(1,3,4,2)(7,11)(10,12)(13,17,18,27)(14,22,23,41)(15,16,28,32)(19,36,26,31)(20,37,38,42)(21,39,30,40)(24,35,25,43)(29,33,44,34)

Polycyclic presentation:

2^9|a^2=e,b^2=gi,c^2=efi,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,g^c=gi,h^d=hi

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, 374015);