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 | |
4111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32)(33,34,40,45)(35,46,37,41)(36,47,48,42)(38,43,44,39);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,35,40,37)(34,41,45,46)(36,44,48,38)(39,47,43,42);(2,6)(3,9)(4,12)(14,16)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31)(33,36,40,48)(34,42,45,47)(35,44,37,38)(39,46,43,41);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31)(33,37,40,35)(34,41,45,46)(36,44,48,38)(39,42,43,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32)(33,38,40,44)(34,43,45,39)(35,36,37,48)(41,42,46,47);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)(33,39,40,43)(34,44,45,38)(35,47,37,42)(36,41,48,46)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=h,d^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8964558);