#6679550

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#35761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1810 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)(11,13)(12,14)(15,16)(17,18,21,24)(19,26,28,23)(20,30,22,25)(27,31,32,29);(1,2,4,5)(3,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,29)(21,28)(22,31)(24,26)(25,27)(30,32);(10,14)(13,16)(17,18)(19,27)(20,25)(21,24)(22,30)(23,31)(26,29)(28,32);(1,3,4,7)(2,6,5,8)(18,24)(23,26)(25,30)(27,32);(2,5)(6,8)(17,20)(18,25)(19,28)(21,22)(24,30)(27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,d^2=fi,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,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6586746);