#313830

order	512
nilpotency class	3
isoclinism	isoclinic class of #264123
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

685 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)(3,6)(4,5)(7,8)(9,10,14,18)(11,16,21,24)(12,22,20,15)(13,23,19,17)(25,26,28,31)(27,33,35,30)(29,32,37,40)(34,39,36,38);(2,5)(6,8)(9,11,13,20,14,21,19,12)(10,15,17,16,18,22,23,24)(25,27,28,35)(26,30,31,33)(29,34,37,36)(32,38,40,39);(1,3,4,7)(2,6,5,8)(9,11,19,12,14,21,13,20)(10,15,23,24,18,22,17,16)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(2,5)(6,8)(9,12,19,21,14,20,13,11)(10,16,23,15,18,24,17,22)(26,31)(30,33)(32,40)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 221026);