#335128

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

603 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=eh,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 242324);