#6693617

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#2657	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1391 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=gi,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,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, 6600813);