#7086609

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×#121	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)(17,18,22,27)(19,24,29,38)(20,25,32,40)(21,23,33,26)(28,37,43,47)(30,35,44,39)(31,41,34,36)(42,48,45,46);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,23)(20,31)(21,33)(24,35)(25,40)(26,27)(28,42)(30,44)(32,34)(37,47)(38,39)(43,45);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,28)(21,30)(22,29)(23,35)(25,37)(26,39)(27,38)(31,42)(32,43)(33,44)(34,45)(36,46)(40,47)(41,48);(11,16)(13,15)(17,20,22,32)(18,25,27,40)(19,28,29,43)(21,34,33,31)(23,36,26,41)(24,37,38,47)(30,45,44,42)(35,46,39,48);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(19,29)(24,38)(28,43)(30,44)(35,39)(37,47)(42,45)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,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=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, 6993805);