#6833517

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 ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1239 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)(10,14)(11,12)(17,18,23,29)(19,30,33,25)(20,34,37,26)(21,39,24,27)(22,28,32,42)(31,41,47,46)(35,43,40,45)(36,44,38,48);(9,10)(11,16)(12,15)(13,14)(17,19,23,33)(18,25,29,30)(20,35,37,40)(21,32,24,22)(26,43,34,45)(27,42,39,28)(31,36,47,38)(41,48,46,44);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,24,38,23,37,21,36)(18,26,27,44,29,34,39,48)(19,31,32,35,33,47,22,40)(25,41,28,43,30,46,42,45);(9,12,13,11)(10,15,14,16)(17,21,23,24)(18,27,29,39)(19,32,33,22)(20,36,37,38)(25,42,30,28)(26,44,34,48)(31,35,47,40)(41,45,46,43);(1,3,4,7)(2,6,5,8)(17,22,21,33,23,32,24,19)(18,28,39,25,29,42,27,30)(20,35,36,31,37,40,38,47)(26,45,48,46,34,43,44,41)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=h,d^2=f,e^2=gh,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, 6740713);