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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4210 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(18,23)(19,25)(24,30)(26,31)(33,34,38,43)(35,41,44,39)(36,40,42,37)(45,46,47,48);(2,6)(7,9)(8,12)(14,15)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(35,44)(39,41)(45,47)(46,48);(2,6)(3,10)(4,13)(14,15)(19,25)(20,27)(21,29)(24,30)(33,35,38,44)(34,39,43,41)(36,47,42,45)(37,48,40,46);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,19,22,25)(18,20,23,27)(21,26,29,31)(24,28,30,32)(33,36,38,42)(34,40,43,37)(35,45,44,47)(39,48,41,46);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20)(18,19)(21,28)(22,27)(23,25)(24,26)(29,32)(30,31)(33,35)(34,41)(36,47)(37,46)(38,44)(39,43)(40,48)(42,45);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,28,27,32)(33,37,38,40)(34,42,43,36)(35,46,44,48)(39,47,41,45)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8900232);