#7273596

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234309
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1265 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=h,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7180792);