#306510

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

559 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)(4,8)(5,7)(9,10,14,18)(11,16,13,15)(12,20,21,23)(17,22,24,19)(25,26,29,32)(27,30,35,39)(28,31,36,40)(33,37,34,38);(1,3,5,4)(2,6,7,8)(10,15)(11,13)(16,18)(17,20)(19,22)(23,24)(25,27)(26,30)(28,33)(29,35)(31,37)(32,39)(34,36)(38,40);(1,4,5,3)(2,6,7,8)(9,11,14,13)(10,16,18,15)(12,19,21,22)(17,23,24,20)(27,33)(28,36)(30,37)(31,40)(34,35)(38,39);(9,12,14,21)(10,17,18,24)(11,19,13,22)(15,20,16,23)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=fh,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,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, 213706);