#6826867

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1488 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)(9,10,13,15)(11,16,12,14)(17,18)(19,24)(20,31)(21,36)(22,27)(23,28)(25,34)(26,37)(29,44)(30,39)(32,42)(33,41)(35,47)(38,46)(40,48)(43,45);(9,10)(11,14)(12,16)(13,15)(17,19,22,30)(18,24,27,39)(20,32,34,48)(21,23,37,33)(25,40,31,42)(26,41,36,28)(29,45,46,35)(38,47,44,43);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,20,33,45,22,34,23,35)(18,25,41,47,27,31,28,43)(19,29,37,48,30,46,21,32)(24,38,26,42,39,44,36,40);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,14,15,16)(20,34)(25,31)(29,46)(32,48)(35,45)(38,44)(40,42)(43,47);(9,12,13,11)(10,14,15,16)(17,21,23,30,22,37,33,19)(18,26,28,24,27,36,41,39)(20,32,35,46,34,48,45,29)(25,42,43,38,31,40,47,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fhi,d^2=gi,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, 6734063);