#8630816

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

4200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8538012);