#6820802

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1295 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,7)(3,6,4,8)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,34)(21,27)(22,37)(23,29)(24,30)(26,31)(28,39)(32,46)(33,42)(35,43)(36,45)(38,48)(40,44)(41,47);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,33)(18,25,29,42)(20,21,31,24)(22,38,39,35)(26,30,34,27)(28,43,37,48)(32,36,47,40)(41,44,46,45);(1,4,5,3)(2,6,7,8)(10,14)(11,16)(17,20,24,19,23,31,21,33)(18,26,30,42,29,34,27,25)(22,35,40,32,39,38,36,47)(28,43,45,46,37,48,44,41);(10,14)(13,15)(17,21,23,24)(18,27,29,30)(19,31,33,20)(22,36,39,40)(25,26,42,34)(28,44,37,45)(32,38,47,35)(41,43,46,48);(17,22,24,40,23,39,21,36)(18,28,30,45,29,37,27,44)(19,32,31,38,33,47,20,35)(25,41,26,43,42,46,34,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6727998);