#6819058

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806575
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×#127	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1402 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=fi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi

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, 6726254);