#6840446

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1349 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)(3,7)(4,8)(5,6)(9,10,13,15)(11,16,12,14)(17,18,22,26)(19,29,31,24)(20,25,34,41)(21,36,37,27)(23,28,33,40)(30,43,45,38)(32,46,47,39)(35,42,44,48);(9,10)(11,14)(12,16)(13,15)(17,19,22,31)(18,24,26,29)(20,32,34,47)(21,23,37,33)(25,39,41,46)(27,28,36,40)(30,44,45,35)(38,48,43,42);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,20,33,44,22,34,23,35)(18,25,40,48,26,41,28,42)(19,30,37,47,31,45,21,32)(24,38,36,46,29,43,27,39);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(18,26)(20,34)(24,29)(27,36)(28,40)(30,45)(32,47)(35,44);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,21,33,19,22,37,23,31)(18,27,40,24,26,36,28,29)(20,32,44,30,34,47,35,45)(25,39,48,38,41,46,42,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fh,d^2=g,e^2=ghi,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, 6747642);