Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4007 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
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)(17,18,23,24)(19,22,27,32)(20,25,30,31)(21,29,28,26);(2,4)(3,10)(8,12)(11,13)(17,19,23,27)(18,22,24,32)(20,21,30,28)(25,29,31,26)(33,34)(35,40)(36,37)(38,39)(41,45)(42,46)(43,48)(44,47);(2,4)(7,9)(8,12)(15,16)(18,24)(19,27)(21,28)(25,31)(34,39)(36,45)(40,47)(42,48);(1,3,6,10)(2,7,4,9)(5,13,14,11)(8,16,12,15)(17,20)(18,25)(19,28)(21,27)(22,26)(23,30)(24,31)(29,32)(33,35)(34,40)(36,42)(37,46)(38,44)(39,47)(41,43)(45,48);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,21,23,28)(18,26,24,29)(19,20,27,30)(22,31,32,25)(33,36)(34,41)(35,42)(37,39)(38,45)(40,43)(44,48)(46,47);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,22)(18,27)(19,24)(20,29)(21,31)(23,32)(25,28)(26,30)(33,37,38,41)(34,36,39,45)(35,43,44,46)(40,48,47,42)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,d^2=g,e^2=h,f^2=hi,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=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8541039);