#312833

order	512
nilpotency class	3
isoclinism	isoclinic class of #264203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

649 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 220029);