#6431036

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1299 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,e^b=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6338232);