#317441

order	512
nilpotency class	3
isoclinism	isoclinic class of #262915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

562 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)(9,10,14,18)(11,15,21,22)(12,16,13,17)(19,24,20,23);(2,5)(6,7)(9,11,13,20,14,21,12,19)(10,15,17,23,18,22,16,24)(25,26,28,32,29,31,37,39)(27,33,34,38,35,40,36,30);(10,16)(11,19)(12,13)(15,22)(17,18)(20,21)(25,27,29,35)(26,30,31,38)(28,36,37,34)(32,40,39,33);(1,3,4,8)(2,6,5,7)(26,31)(30,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dhi,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, 224637);