#6825534

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1495 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,30)(20,34)(21,36)(22,38)(23,29)(24,27)(25,33)(26,31)(28,40)(32,42)(35,43)(37,44)(39,45)(41,48)(46,47);(2,5)(3,7)(17,19,23,33)(18,25,29,30)(20,21,31,24)(22,35,40,39)(26,36,34,27)(28,43,38,45)(32,41,47,37)(42,44,46,48);(10,13)(14,16)(17,20,24,19,23,31,21,33)(18,26,27,25,29,34,36,30)(22,39,41,47,40,35,37,32)(28,45,44,46,38,43,48,42);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21,23,24)(18,27,29,36)(19,31,33,20)(22,37,40,41)(25,26,30,34)(28,44,38,48)(32,39,47,35)(42,43,46,45);(1,3,4,7)(2,6,5,8)(17,22,24,41,23,40,21,37)(18,28,27,44,29,38,36,48)(19,32,31,39,33,47,20,35)(25,42,34,45,30,46,26,43)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=g,e^2=fhi,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,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6732730);