#313821

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, 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

601 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)(9,10,16,22)(11,23,13,18)(12,19,24,17)(14,21,15,20)(25,26,29,32)(27,30,35,39)(28,31,36,40)(33,37,34,38);(2,5)(6,8)(9,11,15,12,16,13,14,24)(10,17,21,18,22,19,20,23)(25,27,29,35)(26,30,32,39)(28,33,36,34)(31,37,40,38);(1,3,4,7)(2,6,5,8)(9,12,15,13,16,24,14,11)(10,18,21,19,22,23,20,17)(27,33)(28,36)(30,37)(31,40)(34,35)(38,39);(2,5)(6,8)(9,13,15,24,16,11,14,12)(10,19,21,23,22,17,20,18)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=eh,d^2=f,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, 221017);