#328732

order	512
nilpotency class	3
isoclinism	isoclinic class of #327021
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

735 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)(11,17)(12,14)(13,21)(15,19)(16,20)(18,24)(22,23)(25,26,30,28)(27,31,34,36)(29,37,39,32)(33,35,40,38);(2,5)(6,8)(10,14)(15,23)(16,21)(17,24)(25,27,30,34)(26,31,28,36)(29,38,39,35)(32,40,37,33);(3,7)(6,8)(9,11)(10,15)(12,18)(13,22)(14,23)(16,24)(17,21)(19,20)(25,28,30,26)(27,33,34,40)(29,32,39,37)(31,38,36,35);(1,3,4,7)(2,6,5,8)(11,18)(15,23)(17,24)(19,22)(27,34)(31,36)(33,40)(35,38)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235928);