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 | |
4180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,10)(4,5)(6,7)(8,11)(9,14)(12,13)(15,16)(17,18,21,24)(19,22,26,25)(20,23,29,28)(27,30,32,31)(33,34)(35,36)(37,45)(38,42)(39,40)(41,48)(43,46)(44,47);(2,7)(3,11)(4,14)(13,15)(17,18)(19,25)(20,23)(21,24)(22,26)(27,31)(28,29)(30,32)(33,35,39,43)(34,36,40,46)(37,47,48,38)(41,42,45,44);(2,7)(4,14)(8,10)(12,16)(19,26)(22,25)(27,32)(30,31)(34,40)(35,43)(37,48)(42,44);(1,3)(2,8)(4,12)(5,15)(6,11)(7,10)(9,13)(14,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,31)(30,32)(33,36)(34,35)(37,44)(38,41)(39,46)(40,43)(42,48)(45,47);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,15)(11,16)(17,19,21,26)(18,22,24,25)(20,27,29,32)(23,30,28,31)(33,37)(34,41)(35,38)(36,44)(39,48)(40,45)(42,46)(43,47);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,16,10,12)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)(33,38)(34,42)(35,37)(36,45)(39,47)(40,44)(41,46)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,e^2=ghi,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=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8528500);