#308022

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

542 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fhi,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, 215218);