#7086619

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1304 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)(3,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,27,30,38)(20,24,33,41)(21,35,36,26)(23,29,39,31)(28,43,32,46)(34,48,47,42)(37,45,40,44);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,25,39)(20,32,33,28)(21,29,36,31)(24,40,41,37)(26,38,35,27)(34,44,47,45)(42,43,48,46);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,28)(21,34)(22,33)(23,37)(25,41)(26,42)(27,43)(29,45)(30,32)(31,44)(35,48)(36,47)(38,46)(39,40);(11,16)(13,15)(19,29)(20,33)(21,36)(23,38)(24,41)(26,35)(27,39)(28,44)(30,31)(32,45)(37,43)(40,46);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(18,25)(19,30)(20,33)(26,35)(29,31)(34,47)(37,40)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6993815);