#6802849

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667652
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 ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1349 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=f,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6710045);