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 | #2416 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3990 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)(3,5,8,13)(4,6,9,15)(11,12,14,16)(19,25)(22,26)(27,31)(30,32)(34,39)(35,41)(36,43)(46,48);(1,3,7,8)(2,5,10,13)(4,11,9,14)(6,12,15,16)(17,18,21,24)(19,26,25,22)(20,23,29,28)(27,32,31,30)(33,34,38,39)(35,37,41,47)(36,44,43,40)(42,48,45,46);(3,8)(5,13)(11,14)(12,16)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,35,38,41)(34,37,39,47)(36,45,43,42)(40,48,44,46);(1,4)(2,6)(3,11)(5,12)(7,9)(8,14)(10,15)(13,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,32)(30,31)(33,34)(35,37)(36,40)(38,39)(41,47)(42,46)(43,44)(45,48);(1,5,7,13)(2,8,10,3)(4,12,9,16)(6,14,15,11)(17,19)(18,22)(20,27)(21,25)(23,30)(24,26)(28,32)(29,31)(33,36,38,43)(34,40,39,44)(35,42,41,45)(37,46,47,48);(1,6)(2,9)(3,12)(4,10)(5,14)(7,15)(8,16)(11,13)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,37)(34,41)(35,39)(36,46)(38,47)(40,45)(42,44)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gi,c^2=h,e^2=hi,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, 8528640);