#6825724

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1314 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=g,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=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, 6732920);