#7086726

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

1299 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)(19,25)(20,26)(21,27)(22,24)(23,28)(29,40)(30,39)(31,36)(32,42)(33,41)(34,38)(35,37)(43,48)(44,46)(45,47);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,24)(19,29)(20,32)(21,22)(25,38)(26,41)(27,28)(30,44)(33,35)(36,40)(43,45)(46,47);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,30)(21,34)(22,31)(23,29)(24,36)(26,39)(27,38)(28,40)(32,45)(33,43)(35,44)(37,46)(41,48)(42,47);(11,16)(13,15)(17,20,23,33)(18,26,28,41)(19,30,29,43)(21,32,22,35)(24,37,27,42)(25,39,40,48)(31,44,34,45)(36,46,38,47);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(17,21,23,22)(18,24,28,27)(19,31,29,34)(20,32,33,35)(25,38,40,36)(26,37,41,42)(30,44,43,45)(39,47,48,46)

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=i,e^2=gi,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=dhi,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, 6993922);