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 | |
4225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,9,4,8)(5,10,13,14)(11,16,12,15)(17,18)(19,21)(20,28)(22,26)(23,25)(24,30)(27,31)(29,32)(33,34)(35,40)(36,37)(38,47)(39,43)(41,46)(42,44)(45,48);(1,3,7,4)(2,8,6,9)(5,12,13,11)(10,16,14,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,29)(30,32)(33,35,39,45)(34,40,43,48)(36,44,46,38)(37,42,41,47);(17,20,22,30)(18,24,26,28)(19,27,25,32)(21,29,23,31)(33,36,39,46)(34,41,43,37)(35,38,45,44)(40,42,48,47);(1,4,7,3)(2,9,6,8)(5,12,13,11)(10,16,14,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28)(33,36)(34,37)(35,44)(38,45)(39,46)(40,42)(41,43)(47,48);(1,5,7,13)(2,10,6,14)(3,11,4,12)(8,15,9,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,31)(30,32)(33,37)(34,36)(35,42)(38,48)(39,41)(40,44)(43,46)(45,47);(1,6,7,2)(3,9,4,8)(5,10,13,14)(11,15,12,16)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,28)(30,31)(33,38,39,44)(34,42,43,47)(35,36,45,46)(37,40,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9770642);