#7083593

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

1470 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)(18,23)(21,33)(24,36)(25,37)(26,34)(29,44)(30,40)(31,35)(38,47)(39,45)(41,48)(43,46);(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,24,28,39)(20,30,32,37)(21,34,33,23)(25,35,40,31)(27,41,42,47)(29,45,44,36)(38,46,48,43);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,27)(21,29)(22,28)(23,36)(25,38)(26,39)(30,41)(31,43)(32,42)(33,44)(34,45)(35,46)(37,47)(40,48);(11,16)(13,15)(17,20,22,32)(18,25,26,40)(19,27,28,42)(21,35,33,31)(23,37,34,30)(24,38,39,48)(29,46,44,43)(36,47,45,41);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,26)(19,28)(23,34)(25,40)(27,42)(29,44)(30,37)(43,46)

Polycyclic presentation:

2^9|b^2=i,c^2=h,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, 6990789);