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 | |
4168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18,24,29)(19,27,21,25)(20,26,22,31)(23,28,32,30)(33,34,39,44)(35,40,47,46)(36,43,38,41)(37,42,45,48);(1,2)(3,10)(4,12)(5,6)(7,9)(8,13)(11,14)(15,16)(17,19,24,21)(18,25,29,27)(20,30,22,28)(23,26,32,31)(33,35)(34,40)(36,37)(38,45)(39,47)(41,48)(42,43)(44,46);(2,6)(7,10)(8,12)(14,16)(17,20,24,22)(18,26,29,31)(19,30,21,28)(23,25,32,27)(33,36,39,38)(34,41,44,43)(35,45,47,37)(40,42,46,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21,24,19)(18,27,29,25)(20,30,22,28)(23,26,32,31)(33,37,39,45)(34,42,44,48)(35,36,47,38)(40,43,46,41);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,22,24,20)(18,26,29,31)(19,30,21,28)(23,27,32,25)(33,38,39,36)(34,43,44,41)(35,45,47,37)(40,42,46,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,23,24,32)(18,28,29,30)(19,31,21,26)(20,25,22,27)(33,34)(35,46)(36,41)(37,42)(38,43)(39,44)(40,47)(45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gh,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662981);