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 | |
4124 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18)(19,20)(21,29)(22,23)(24,31)(25,28)(26,32)(27,30)(33,34,38,37)(35,41,43,39)(36,44,46,40)(42,47,45,48);(2,3)(5,13)(7,9)(12,16)(17,19,22,25)(18,20,23,28)(21,30,31,26)(24,32,29,27)(33,35,38,43)(34,39,37,41)(36,45,46,42)(40,48,44,47);(4,11)(7,9)(12,16)(14,15)(18,23)(19,25)(24,29)(26,30)(34,37)(39,41)(40,44)(47,48);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,30)(31,32)(33,36,38,46)(34,40,37,44)(35,42,43,45)(39,47,41,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)(33,37,38,34)(35,41,43,39)(36,44,46,40)(42,48,45,47);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,24)(19,26)(20,27)(22,31)(23,29)(25,30)(28,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,48)(45,46)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10231817);