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 | |
4173 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)(8,15)(10,13)(11,16)(17,18,22,23)(19,25,27,20)(21,29,30,24)(26,28,32,31)(33,34,39,44)(35,43,46,40)(36,41,47,48)(37,42,45,38);(1,2,6,7)(3,8,9,15)(4,12,14,5)(10,11,13,16)(18,23)(20,25)(24,29)(28,31)(33,35,39,46)(34,40,44,43)(36,45,47,37)(38,41,42,48);(2,7)(5,12)(8,15)(11,16)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(33,36,39,47)(34,41,44,48)(35,45,46,37)(38,40,42,43);(1,3)(2,8)(4,13)(5,11)(6,9)(7,15)(10,14)(12,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,37)(34,42)(35,36)(38,44)(39,45)(40,48)(41,43)(46,47);(1,4)(2,5)(3,10)(6,14)(7,12)(8,16)(9,13)(11,15)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,35)(34,43)(36,45)(37,47)(38,41)(39,46)(40,44)(42,48);(1,5)(2,4)(3,11)(6,12)(7,14)(8,13)(9,16)(10,15)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)(33,38,39,42)(34,37,44,45)(35,41,46,48)(36,43,47,40)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833433);