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 | |
4238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,5,8,11)(6,9,15,12)(10,13,14,16)(17,18,21,25)(19,23,27,22)(20,28,29,24)(26,30,32,31)(33,34,39,44)(35,45,46,40)(36,37,42,41)(38,43,47,48);(1,3,7,8)(2,5,4,11)(6,14,15,10)(9,16,12,13)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,35)(34,40)(36,47)(37,43)(38,42)(39,46)(41,48)(44,45);(3,8)(5,11)(10,14)(13,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,29)(30,32)(33,34,39,44)(35,40,46,45)(36,41,42,37)(38,43,47,48);(1,4,7,2)(3,5,8,11)(6,12,15,9)(10,13,14,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,36)(34,41)(35,38)(37,44)(39,42)(40,43)(45,48)(46,47);(1,5)(2,8)(3,4)(6,13)(7,11)(9,14)(10,12)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,37,39,41)(34,42,44,36)(35,43,46,48)(38,45,47,40);(1,6)(2,9)(3,10)(4,12)(5,13)(7,15)(8,14)(11,16)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)(33,38)(34,43)(35,36)(37,45)(39,47)(40,41)(42,46)(44,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=hi,e^2=gh,f^2=h,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, 10268126);