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 | |
4268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(3,9)(4,11)(15,16)(17,18,21,25)(19,23,27,22)(20,28,29,24)(26,30,32,31)(33,34,39,44)(35,42,37,40)(36,41,43,38)(45,46,47,48);(1,2,5,6)(3,7,9,13)(4,8,11,14)(10,15,12,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(35,37)(40,42)(45,47)(46,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,15)(11,12)(14,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,29)(30,32)(33,35,39,37)(34,40,44,42)(36,45,43,47)(38,46,41,48);(1,3)(2,7)(4,12)(5,9)(6,13)(8,16)(10,11)(14,15)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,36)(34,41)(35,45)(37,47)(38,44)(39,43)(40,48)(42,46);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,14)(12,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,37,39,35)(34,42,44,40)(36,47,43,45)(38,48,41,46);(1,4,5,11)(2,8,6,14)(3,10,9,12)(7,15,13,16)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)(33,38,39,41)(34,43,44,36)(35,46,37,48)(40,47,42,45)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10329545);