#316835

order	512
nilpotency class	3
isoclinism	isoclinic class of #262851
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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,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=dh,f^c=fh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224031);