#7083605

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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1491 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)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,29)(19,30,34,46)(20,35,21,26)(22,28,23,27)(25,39,42,33)(31,47,36,41)(32,40,48,43)(37,44,38,45);(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,36)(21,31)(22,33)(23,39)(24,34)(26,43)(27,30)(28,46)(29,42)(32,38)(35,40)(37,48)(41,44)(45,47);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,26)(19,31)(21,24)(22,38)(23,37)(25,40)(27,45)(28,44)(29,35)(30,41)(32,39)(33,48)(34,36)(42,43)(46,47);(11,16)(13,15)(17,21,24,20)(18,26,29,35)(19,32,34,48)(22,38,23,37)(25,41,42,47)(27,44,28,45)(30,40,46,43)(31,33,36,39);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,22,24,23)(18,27,29,28)(19,33,34,39)(20,37,21,38)(25,30,42,46)(26,44,35,45)(31,32,36,48)(40,47,43,41)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=i,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=ehi,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, 6990801);