#323328

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
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×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

608 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)(3,7,4,6)(9,10,12,16,13,15,21,23)(11,17,18,22,19,24,20,14)(25,26,29,32)(27,30,35,39)(28,31,36,40)(33,37,34,38);(1,3,5,4)(2,6,8,7)(25,27,28,34,29,35,36,33)(26,30,31,38,32,39,40,37);(9,11,13,19)(10,14,15,22)(12,20,21,18)(16,24,23,17)(27,33)(28,36)(30,37)(31,40)(34,35)(38,39);(1,4,5,3)(2,7,8,6)(10,15)(14,22)(16,23)(17,24)

Polycyclic presentation:

2^9|a^2=efi,b^2=eg,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230524);