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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,13,6)(4,14,5,9)(11,15,12,16)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34,40,44)(35,37,38,41)(36,45,46,42)(39,43,48,47);(1,3,7,13)(2,8,10,6)(4,12,5,11)(9,15,14,16)(19,26)(23,30)(27,31)(28,32)(33,35)(34,41)(36,43)(37,44)(38,40)(39,45)(42,48)(46,47);(1,4,7,5)(2,9,10,14)(3,11,13,12)(6,15,8,16)(17,19,22,26)(18,23,25,30)(20,27,24,31)(21,28,29,32)(33,36,40,46)(34,42,44,45)(35,43,38,47)(37,39,41,48);(1,5,7,4)(2,9,10,14)(3,12,13,11)(6,15,8,16)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,37)(34,35)(36,39)(38,44)(40,41)(42,43)(45,47)(46,48);(1,2)(3,6)(4,14)(5,9)(7,10)(8,13)(11,16)(12,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,38,40,35)(34,37,44,41)(36,43,46,47)(39,42,48,45);(1,6)(2,3)(4,15)(5,16)(7,8)(9,11)(10,13)(12,14)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,39)(34,43)(35,45)(36,41)(37,46)(38,42)(40,48)(44,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8509822);