#8630828

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

4189 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=h,e^2=i,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, 8538024);