#320805

order	512
nilpotency class	3
isoclinism	isoclinic class of #298939
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

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

656 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)(3,7)(4,5)(6,8)(9,10,14,16,13,18,12,17)(11,15,21,22,19,23,20,24);(2,5)(6,7)(9,11)(10,15)(12,20)(13,19)(14,21)(16,22)(17,24)(18,23)(25,26,28,31,29,32,37,40)(27,33,34,38,35,39,36,30);(10,16)(12,14)(15,22)(17,18)(20,21)(23,24)(25,27,29,35)(26,30,32,38)(28,36,37,34)(31,39,40,33);(1,3,4,8)(2,6,5,7)(11,19)(15,23)(20,21)(22,24)

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=i,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,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, 228001);