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 | #50094 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4060 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)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,34)(35,40)(36,38)(37,42)(39,41)(43,48)(44,47)(45,46);(2,3)(4,11)(8,10)(12,16)(17,19,22,27)(18,23,21,25)(20,26,28,29)(24,31,30,32)(33,35,39,37)(34,40,41,42)(36,46,43,44)(38,45,48,47);(4,11)(5,13)(7,9)(8,10)(18,21)(20,28)(23,25)(26,29)(34,41)(35,37)(38,48)(44,46);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,23)(20,29)(21,25)(22,27)(24,32)(26,28)(30,31)(33,36,39,43)(34,38,41,48)(35,44,37,46)(40,47,42,45);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20,22,28)(18,24,21,30)(19,26,27,29)(23,31,25,32)(33,37,39,35)(34,42,41,40)(36,44,43,46)(38,47,48,45);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,18)(19,25,27,23)(20,30,28,24)(26,32,29,31)(33,38)(34,43)(35,45)(36,41)(37,47)(39,48)(40,44)(42,46)Polycyclic presentation:
2^9|b^2=h,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8947941);