#8726363

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=gh,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633559);