#464711

order	512
nilpotency class	3
isoclinism	isoclinic class of #454399
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371907);