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 | |
4508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,11,16)(4,14,12,6)(5,9,15,13)(17,18)(19,25)(20,24)(21,27)(22,23)(26,31)(28,30)(29,32)(33,34,38,39)(35,40,44,48)(36,37,41,47)(42,46,45,43);(1,3)(2,8)(4,5)(6,13)(7,11)(9,14)(10,16)(12,15)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,31)(30,32)(34,39)(36,41)(40,48)(42,45);(3,11)(4,12)(6,14)(8,16)(18,23)(21,25)(24,28)(29,31)(33,35,38,44)(34,40,39,48)(36,42,41,45)(37,46,47,43);(1,4)(2,6)(3,5)(7,12)(8,13)(9,16)(10,14)(11,15)(17,18)(19,21)(20,28)(22,23)(24,30)(25,27)(26,31)(29,32)(33,35)(34,40)(36,45)(37,43)(38,44)(39,48)(41,42)(46,47);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,13)(14,16)(17,20,22,30)(18,24,23,28)(19,26,27,32)(21,29,25,31)(33,36,38,41)(34,37,39,47)(35,42,44,45)(40,46,48,43);(1,6)(2,4)(3,13)(5,8)(7,14)(9,11)(10,12)(15,16)(17,21)(18,19)(20,29)(22,25)(23,27)(24,26)(28,32)(30,31)(33,37)(34,41)(35,43)(36,39)(38,47)(40,42)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=ghi,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8835877);