#310131

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

582 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)(3,6)(4,5)(7,8)(9,10,12,16,13,17,22,15)(11,18,19,23,20,24,21,14)(25,26)(27,33)(28,30)(29,32)(31,35)(34,39)(36,38)(37,40);(9,11,13,20)(10,14,17,23)(12,21,22,19)(15,18,16,24)(26,30)(27,34)(29,36)(31,39)(33,40)(35,37);(2,5)(6,8)(25,27,28,35)(26,31,30,33)(29,37,36,34)(32,39,38,40);(1,3,4,7)(2,6,5,8)(10,15)(12,22)(14,18)(16,17)(19,21)(23,24)(26,30)(31,33)(32,38)(39,40)

Polycyclic presentation:

2^9|a^2=ei,b^2=h,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,g^b=gi,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 217327);