#6830265

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	#42926	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1548 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,28)(20,22)(21,30)(23,27)(24,26)(25,29)(31,32);(9,11)(10,13)(12,16)(14,15)(17,19)(18,25)(20,21)(22,26)(23,29)(24,31)(27,28)(30,32);(2,5)(6,8)(10,14)(11,16)(17,20,21,19,23,31,24,29)(18,22,30,28,27,32,26,25);(10,14)(13,15)(17,21,23,24)(18,26,27,30)(19,20,29,31)(22,28,32,25);(1,3,4,7)(2,6,5,8)(17,22,21,28,23,32,24,25)(18,20,30,19,27,31,26,29)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6737461);