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 | |
4217 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)(3,10)(4,9)(5,7)(6,8)(11,16)(12,15)(13,14)(17,18,22,25)(19,23,28,31)(20,21,29,24)(26,27,32,30)(33,34,38,42)(35,39,44,48)(36,45,46,40)(37,41,43,47);(1,3,6,5)(2,7,8,10)(4,11,12,13)(9,14,15,16)(17,19,22,28)(18,23,25,31)(20,26,29,32)(21,27,24,30)(33,35)(34,39)(36,37)(38,44)(40,47)(41,45)(42,48)(43,46);(2,8)(7,10)(9,15)(14,16)(17,20,22,29)(18,24,25,21)(19,26,28,32)(23,30,31,27)(33,36,38,46)(34,40,42,45)(35,43,44,37)(39,41,48,47);(1,2)(3,7)(4,9)(5,10)(6,8)(11,14)(12,15)(13,16)(17,20)(18,21)(19,26)(22,29)(23,27)(24,25)(28,32)(30,31)(33,36)(34,40)(35,37)(38,46)(39,47)(41,48)(42,45)(43,44);(1,4)(2,9)(3,11)(5,13)(6,12)(7,14)(8,15)(10,16)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37)(34,41)(35,36)(38,43)(39,45)(40,48)(42,47)(44,46);(1,5,6,3)(2,10,8,7)(4,11,12,13)(9,14,15,16)(17,21)(18,20)(19,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,43)(44,48)(45,46)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9820493);