#7086981

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

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=dfi,e^b=eh,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, 6994177);