#6819707

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1351 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)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15);(1,3,5,4)(2,6,7,8)(9,11,14,10)(12,16,13,15)(18,22)(19,25)(20,26)(23,28)(24,30)(27,31);(1,4,5,3)(2,6,7,8)(9,12,14,13)(10,15,11,16)(17,18,20,22,21,23,26,28)(19,24,25,29,27,30,31,32);(9,12,14,13)(10,15,11,16)(18,23)(22,28)(24,30)(29,32);(9,13,14,12)(10,15,11,16)(17,19,26,31,21,27,20,25)(18,24,28,32,23,30,22,29)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=f,e^2=fhi,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, 6726903);