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 |
| prime radical | #51130 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4048 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)(3,10)(4,8)(5,14)(6,12)(7,9)(11,15)(13,16)(17,18,22,24)(19,25,27,20)(21,23,29,30)(26,31,32,28);(1,3,7,4)(2,8,9,10)(5,13,15,12)(6,11,16,14)(17,18)(19,25)(20,27)(21,23)(22,24)(26,31)(28,32)(29,30)(33,34)(35,41)(36,44)(37,38)(39,43)(40,46)(42,47)(45,48);(2,9)(6,16)(8,10)(11,14)(21,29)(23,30)(26,32)(28,31)(34,38)(39,41)(40,44)(47,48);(1,4,7,3)(2,10,9,8)(5,12,15,13)(6,14,16,11)(17,19,22,27)(18,20,24,25)(21,26,29,32)(23,28,30,31)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,46)(44,47);(1,5,7,15)(2,11,9,14)(3,12,4,13)(6,10,16,8)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,36)(34,40)(35,42)(37,46)(38,44)(39,47)(41,48)(43,45);(1,6,7,16)(2,12,9,13)(3,11,4,14)(5,10,15,8)(17,21,22,29)(18,23,24,30)(19,26,27,32)(20,28,25,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8292129);