#6912728

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1609 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^d=ei

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, 6819924);