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 | |
4099 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)(3,12)(4,8)(5,9)(6,13)(7,11)(10,16)(14,15)(17,18,21,25)(19,23,27,22)(20,28,29,24)(26,30,32,31)(33,34)(35,41)(36,44)(37,40)(38,43)(39,46)(42,47)(45,48);(3,4)(6,16)(8,12)(10,13)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,3,7,4)(2,8,11,12)(5,10,15,13)(6,14,16,9)(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,29)(30,32)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(1,4,7,3)(2,8,11,12)(5,10,15,13)(6,9,16,14)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,35)(34,38)(36,42)(37,43)(39,47)(40,41)(44,48)(45,46);(1,5,7,15)(2,9,11,14)(3,13,4,10)(6,8,16,12)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,47)(44,46)(45,48);(1,6,7,16)(2,10,11,13)(3,14,4,9)(5,8,15,12)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)(33,36,37,46)(34,39,40,44)(35,42,43,45)(38,47,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=gh,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, 10256234);