#6822385

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1300 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)(10,14)(11,12)(17,18,23,28)(19,24,31,40)(20,32,37,25)(21,26,34,42)(22,27,30,39)(29,45,47,38)(33,44,36,41)(35,48,46,43);(9,10)(11,15)(12,16)(13,14)(17,19,23,31)(18,24,28,40)(20,33,37,36)(21,30,34,22)(25,41,32,44)(26,39,42,27)(29,46,47,35)(38,48,45,43);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,34,46,23,37,21,35)(18,25,42,48,28,32,26,43)(19,29,30,36,31,47,22,33)(24,38,39,44,40,45,27,41);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,23,34)(18,26,28,42)(19,30,31,22)(20,35,37,46)(24,39,40,27)(25,43,32,48)(29,36,47,33)(38,44,45,41);(10,14)(15,16)(17,22,21,31,23,30,34,19)(18,27,26,40,28,39,42,24)(20,36,35,29,37,33,46,47)(25,44,43,38,32,41,48,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fhi,d^2=g,e^2=h,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, 6729581);