#7089446

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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

1286 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,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

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, 6996642);