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 | |
4315 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(4,11)(8,10)(12,16)(17,18,22,23)(19,20,27,24)(21,29,31,25)(26,32,30,28)(33,34)(35,42)(36,46)(37,40)(38,47)(39,41)(43,48)(44,45);(1,2,6,3)(4,9,11,7)(5,8,13,10)(12,15,16,14)(18,23)(20,24)(25,29)(28,32)(33,35,39,37)(34,40,41,42)(36,44,47,43)(38,45,46,48);(17,19)(18,24)(20,23)(21,26)(22,27)(25,32)(28,29)(30,31)(33,36)(34,38)(35,43)(37,44)(39,47)(40,48)(41,46)(42,45);(1,3,6,2)(4,7,11,9)(5,8,13,10)(12,15,16,14)(17,19)(18,24)(20,23)(21,30)(22,27)(25,28)(26,31)(29,32)(33,36)(34,38)(35,44)(37,43)(39,47)(40,45)(41,46)(42,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,20,22,24)(18,19,23,27)(21,28,31,32)(25,26,29,30)(33,37,39,35)(34,40,41,42)(36,44,47,43)(38,48,46,45);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,25)(19,26)(20,28)(22,31)(23,29)(24,32)(27,30)(33,38)(34,36)(35,45)(37,48)(39,46)(40,44)(41,47)(42,43)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10327135);