#7086674

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1279 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)(17,18,23,28)(19,25,31,40)(20,26,33,41)(21,27,22,24)(29,38,45,47)(30,36,34,39)(32,42,35,37)(43,48,44,46);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,24)(20,32)(21,22)(25,36)(26,41)(27,28)(29,43)(30,34)(33,35)(38,47)(39,40)(44,45);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,29)(21,34)(22,30)(23,31)(24,36)(26,38)(27,39)(28,40)(32,43)(33,45)(35,44)(37,46)(41,47)(42,48);(11,16)(13,15)(17,20,23,33)(18,26,28,41)(19,29,31,45)(21,32,22,35)(24,37,27,42)(25,38,40,47)(30,44,34,43)(36,46,39,48);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(17,21,23,22)(18,27,28,24)(19,30,31,34)(20,32,33,35)(25,36,40,39)(26,42,41,37)(29,44,45,43)(38,46,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,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=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, 6993870);