#6993560

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885065
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1371 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)(3,7)(4,6)(5,8)(9,10)(11,19)(12,13)(14,17)(15,18)(16,22)(20,23)(21,24)(27,33)(28,35)(30,37)(32,39)(34,36)(38,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,24)(18,19)(25,26,29,31)(27,30,34,38)(28,32,35,39)(33,37,36,40);(3,8)(5,7)(9,12,15,17)(10,13,18,14)(11,20,22,24)(16,21,19,23)(25,27,29,34)(26,30,31,38)(28,36,35,33)(32,40,39,37);(2,6)(5,7)(9,13,15,14)(10,17,18,12)(11,21,22,23)(16,24,19,20)(26,31)(30,38)(32,39)(37,40);(9,14,15,13)(10,17,18,12)(11,21,22,23)(16,20,19,24)

Polycyclic presentation:

2^9|b^2=hi,c^2=fhi,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,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, 6900756);