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 | |
4145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,22,23)(19,24,28,27)(20,25,29,32)(21,30,31,26)(33,34,38,42)(35,43,44,39)(36,40,46,47)(37,45,48,41);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,15)(14,16)(17,18,22,23)(19,27,28,24)(20,25,29,32)(21,26,31,30)(33,35,38,44)(34,39,42,43)(36,41,46,45)(37,40,48,47);(2,6)(3,10)(4,13)(15,16)(18,23)(19,28)(20,29)(21,31)(33,35,38,44)(34,39,42,43)(36,45,46,41)(37,47,48,40);(1,3)(2,7)(4,14)(5,10)(6,9)(8,16)(11,13)(12,15)(17,19,22,28)(18,24,23,27)(20,30,29,26)(21,25,31,32)(33,35)(34,39)(36,45)(37,40)(38,44)(41,46)(42,43)(47,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,15)(9,16)(10,14)(17,20)(18,25)(19,26)(21,24)(22,29)(23,32)(27,31)(28,30)(33,36,38,46)(34,40,42,47)(35,41,44,45)(37,39,48,43);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,21)(18,26)(19,25)(20,24)(22,31)(23,30)(27,29)(28,32)(33,37,38,48)(34,41,42,45)(35,40,44,47)(36,39,46,43)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=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, 8965722);