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 | |
4146 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)(17,18,23,24)(19,22,27,30)(20,25,29,31)(21,26,28,32)(33,34)(35,43)(36,45)(37,42)(38,40)(39,46)(41,47)(44,48);(2,3)(5,13)(7,9)(12,16)(17,19,23,27)(18,22,24,30)(20,28,29,21)(25,32,31,26)(33,35,38,46)(34,39,40,43)(36,42,47,44)(37,45,48,41);(4,11)(7,9)(12,16)(14,15)(18,24)(19,27)(21,28)(25,31)(34,40)(37,48)(39,43)(41,45);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20)(18,25)(19,21)(22,26)(23,29)(24,31)(27,28)(30,32)(33,36,38,47)(34,41,40,45)(35,44,46,42)(37,39,48,43);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21)(18,26)(19,20)(22,25)(23,28)(24,32)(27,29)(30,31)(33,37,38,48)(34,42,40,44)(35,45,46,41)(36,39,47,43);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,19)(20,26)(21,25)(23,30)(24,27)(28,31)(29,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,43)(41,48)(42,47)Polycyclic presentation:
2^9|a^2=i,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, 10231755);