#6840524

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fhi,d^2=gi,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,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, 6747720);