#323797

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

584 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)(9,10,14,16,13,18,12,17)(11,15,21,22,19,23,20,24)(25,26,28,32,29,33,35,39)(27,34,38,40,37,30,36,31);(1,3,5,4)(2,6,8,7)(9,11)(10,15)(12,20)(13,19)(14,21)(16,22)(17,24)(18,23)(25,27,35,36,29,37,28,38)(26,30,39,40,33,34,32,31);(10,16)(12,14)(15,22)(17,18)(20,21)(23,24)(25,27,29,37)(26,31,33,40)(28,36,35,38)(30,39,34,32);(1,4,5,3)(2,7,8,6)(11,19)(15,23)(20,21)(22,24)(25,28,29,35)(26,32,33,39)(27,36,37,38)(30,40,34,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230993);