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 | |
4096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,9)(8,12)(14,15)(17,18,22,24)(19,26,21,23)(20,28,30,25)(27,31,29,32)(33,34)(35,40)(36,37)(38,43)(39,41)(42,47)(44,48)(45,46);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19,22,21)(18,23,24,26)(20,29,30,27)(25,32,28,31)(33,35,39,44)(34,40,41,48)(36,45,42,43)(37,46,47,38);(3,10)(7,9)(11,16)(14,15)(18,24)(23,26)(25,28)(31,32)(34,41)(35,44)(37,47)(43,45);(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,20,22,30)(18,25,24,28)(19,27,21,29)(23,31,26,32)(33,36,39,42)(34,37,41,47)(35,43,44,45)(38,48,46,40);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18)(19,26)(20,25)(21,23)(22,24)(27,32)(28,30)(29,31)(33,37)(34,42)(35,38)(36,41)(39,47)(40,45)(43,48)(44,46);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21,22,19)(18,23,24,26)(20,27,30,29)(25,32,28,31)(33,38)(34,43)(35,37)(36,40)(39,46)(41,45)(42,48)(44,47)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=h,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, 8495080);