#6700332

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1807 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=fi,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6607528);