#341684

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

494 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 248880);