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 | #2617 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(17,18,22,26)(19,27,28,23)(20,24,30,29)(21,25,32,31)(33,34,40,43)(35,39,46,41)(36,38,37,42)(44,47,48,45);(1,2)(3,9)(4,12)(5,6)(7,11)(8,13)(10,15)(14,16)(17,18)(19,27)(20,29)(21,25)(22,26)(23,28)(24,30)(31,32)(33,35,40,46)(34,41,43,39)(36,44,37,48)(38,45,42,47);(2,6)(4,13)(7,9)(10,14)(20,30)(21,32)(24,29)(25,31)(33,36,40,37)(34,42,43,38)(35,44,46,48)(39,45,41,47);(1,3)(2,7)(4,14)(5,11)(6,9)(8,16)(10,13)(12,15)(17,19,22,28)(18,23,26,27)(20,31,30,25)(21,24,32,29)(33,37,40,36)(34,38,43,42)(35,44,46,48)(39,45,41,47);(1,4)(2,8)(3,10)(5,13)(6,12)(7,15)(9,16)(11,14)(17,20)(18,24)(19,25)(21,23)(22,30)(26,29)(27,32)(28,31)(33,38)(34,37)(35,45)(36,43)(39,44)(40,42)(41,48)(46,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,13)(14,16)(17,21)(18,25)(19,24)(20,23)(22,32)(26,31)(27,30)(28,29)(33,39)(34,35)(36,47)(37,45)(38,44)(40,41)(42,48)(43,46)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^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, 8947536);