#6844008

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1297 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,23,26)(19,28,29,22)(20,31,30,24)(21,27,32,25);(9,11)(10,13)(12,16)(14,15)(17,19,23,29)(18,22,26,28)(20,32,30,21)(24,25,31,27);(2,6)(7,8)(11,16)(13,15)(17,20,21,19,23,30,32,29)(18,24,27,22,26,31,25,28);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,21,23,32)(18,25,26,27)(19,20,29,30)(22,31,28,24);(1,4,5,3)(2,8,6,7)(17,22,32,24,23,28,21,31)(18,19,25,20,26,29,27,30)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fhi,d^2=g,e^2=fghi,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, 6751204);