#6993430

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1425 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=fhi,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6900626);