#6840390

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1357 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)(19,24)(20,25)(21,23)(22,26)(27,40)(28,37)(29,35)(30,39)(31,45)(32,42)(33,36)(34,41)(38,43)(44,48)(46,47);(9,10)(11,14)(12,16)(13,15)(18,23)(19,27)(20,31)(22,28)(24,38)(25,41)(29,43)(33,34)(35,40)(36,45)(39,42)(44,46);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,28,43,21,29,22,27)(18,24,37,38,23,35,26,40)(20,32,31,44,33,30,34,46)(25,42,45,48,36,39,41,47);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(18,23)(19,29)(25,36)(26,37)(27,43)(30,32)(41,45)(44,46);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,20,28,31,21,33,22,34)(18,25,37,45,23,36,26,41)(19,30,43,46,29,32,27,44)(24,39,38,47,35,42,40,48)

Polycyclic presentation:

2^9|a^2=fi,c^2=fh,d^2=g,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6747586);