#7086789

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1324 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)(19,24)(20,25)(21,23)(22,26)(27,33)(28,39)(29,38)(30,35)(31,41)(32,40)(34,36)(37,43)(42,48)(44,46)(45,47);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(20,31)(21,33)(24,37)(25,40)(26,27)(29,44)(32,34)(35,39)(42,45)(46,47);(2,7)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,29)(21,30)(22,28)(23,35)(25,38)(26,39)(27,37)(31,45)(32,42)(33,43)(34,44)(36,46)(40,48)(41,47);(11,16)(13,15)(17,20,22,32)(18,25,26,40)(19,29,28,42)(21,34,33,31)(23,36,27,41)(24,38,39,48)(30,44,43,45)(35,46,37,47);(1,4,5,3)(2,8,7,6)(10,14)(13,15)(18,26)(19,28)(23,27)(25,40)(29,42)(30,43)(36,41)(44,45)

Polycyclic presentation:

2^9|b^2=g,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=dhi,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6993985);