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 | #2464 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3942 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,22,25)(19,26,21,23)(20,24,29,32)(27,31,28,30)(33,34,40,44)(35,45,37,41)(36,42,46,47)(38,39,43,48);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,35,40,37)(34,41,44,45)(36,38,46,43)(39,42,48,47);(3,10)(6,7)(11,16)(14,15)(17,19,22,21)(18,23,25,26)(20,28,29,27)(24,31,32,30)(33,36,40,46)(34,42,44,47)(35,38,37,43)(39,41,48,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20,22,29)(18,24,25,32)(19,27,21,28)(23,30,26,31)(33,37,40,35)(34,41,44,45)(36,38,46,43)(39,47,48,42);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21,22,19)(18,23,25,26)(20,27,29,28)(24,31,32,30)(33,38)(34,39)(35,46)(36,37)(40,43)(41,42)(44,48)(45,47);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,39)(34,43)(35,42)(36,41)(37,47)(38,44)(40,48)(45,46)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9074778);