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 |
| prime radical | #52846 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,12)(4,14)(5,6)(7,9)(8,13)(10,16)(11,15)(33,34,39,40)(35,36,43,48)(37,41,38,42)(44,46,45,47);(1,3,7,13)(2,8,9,12)(4,15,16,5)(6,10,11,14)(17,18,22,23)(19,21,25,30)(20,24,29,31)(26,28,27,32)(34,40)(35,43)(41,42)(44,45);(2,9)(6,11)(8,12)(10,14)(18,23)(21,30)(24,31)(28,32)(33,35)(34,36)(37,44)(38,45)(39,43)(40,48)(41,46)(42,47);(1,4,7,16)(2,10,9,14)(3,5,13,15)(6,12,11,8)(17,19,22,25)(18,21,23,30)(20,27,29,26)(24,32,31,28)(33,36)(34,35)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45);(1,5)(2,11)(3,4)(6,9)(7,15)(8,14)(10,12)(13,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,27)(30,32)(33,37,39,38)(34,41,40,42)(35,44,43,45)(36,46,48,47);(1,6,7,11)(2,5,9,15)(3,10,13,14)(4,12,16,8)(17,21)(18,25)(19,23)(20,28)(22,30)(24,27)(26,31)(29,32)(33,38,39,37)(34,42,40,41)(35,45,43,44)(36,47,48,46)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8202772);