#345512

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
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

543 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:

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=ei,c^2=eh,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defhi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 252708);