#325862

order	512
nilpotency class	3
isoclinism	isoclinic class of #288755
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

498 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)(11,19)(12,15)(13,23)(14,18)(16,21)(17,20)(22,24)(25,26,29,32)(27,30,36,40)(28,31,33,37)(34,38,35,39);(9,11,20,22,14,21,13,12)(10,15,23,16,18,24,17,19)(25,27,33,34,29,36,28,35)(26,30,37,38,32,40,31,39);(1,3,4,8)(2,5,6,7)(11,21)(12,22)(15,24)(16,19)(27,34)(28,33)(30,38)(31,37)(35,36)(39,40);(9,12,13,21,14,22,20,11)(10,16,17,15,18,19,23,24)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233058);