#310104

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

566 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)(9,10,15,18,14,20,13,19)(11,17,23,16,12,22,21,24)(27,33)(30,39)(34,40)(35,38);(9,11,14,12)(10,16,20,24)(13,23,15,21)(17,19,22,18)(25,26,28,31)(27,34,33,40)(29,37,36,32)(30,38,39,35);(2,5)(6,8)(9,12,14,11)(10,17,20,22)(13,23,15,21)(16,19,24,18)(25,27,28,33)(26,30,31,39)(29,38,36,35)(32,40,37,34);(1,3,4,7)(2,6,5,8)(10,18)(13,15)(16,17)(19,20)(21,23)(22,24)

Polycyclic presentation:

2^9|a^2=eh,b^2=hi,c^2=hi,d^2=fhi,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, 217300);