#7086430

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

1380 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)(9,10)(11,13)(12,14)(15,16)(17,18,21,27,22,26,35,41)(19,28,31,40,30,24,38,23)(20,32,37,46,36,25,43,45)(29,42,33,39,34,44,47,48);(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,26,40)(20,33,36,47)(21,38,35,31)(24,41,28,27)(25,44,32,42)(29,43,34,37)(39,46,48,45);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,34)(21,31)(22,30)(23,27)(25,44)(26,28)(29,36)(32,42)(33,43)(35,38)(37,47)(39,46)(40,41)(45,48);(11,16)(13,15)(17,20,35,43,22,36,21,37)(18,25,41,46,26,32,27,45)(19,29,38,47,30,34,31,33)(23,39,24,42,40,48,28,44);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,26)(19,30)(20,36)(27,41)(31,38)(37,43)(39,48)(42,44)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=hi,d^2=fi,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6993626);