#6909720

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1791 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=fi,e^2=i,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, 6816916);