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)(7,14)(10,12)(15,16)(17,18,24,29)(19,30,32,25)(20,26,22,28)(21,27,31,23)(33,34)(35,39)(36,40)(37,45)(38,42)(41,46)(43,44)(47,48);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,19,24,32)(18,25,29,30)(20,31,22,21)(23,26,27,28)(33,35)(34,39)(36,42)(37,43)(38,40)(41,47)(44,45)(46,48);(2,6)(7,14)(8,11)(15,16)(17,20,24,22)(18,26,29,28)(19,31,32,21)(23,25,27,30)(33,34,38,42)(35,36,40,39)(37,45,46,41)(43,47,48,44);(1,3)(2,7)(4,12)(5,9)(6,14)(8,16)(10,13)(11,15)(17,21)(18,27)(19,20)(22,32)(23,29)(24,31)(25,28)(26,30)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,47)(46,48);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,22,24,20)(18,28,29,26)(19,31,32,21)(23,25,27,30)(33,36)(34,40)(35,42)(37,44)(38,39)(41,48)(43,45)(46,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,23,24,27)(18,21,29,31)(19,26,32,28)(20,30,22,25)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,e^2=g,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, 8832198);