Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #55435 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4061 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)(3,11,15,8)(4,5,9,6)(12,14,16,13)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32)(33,34,37,39)(35,38,41,46)(36,40,44,47)(42,45,43,48);(1,3,7,15)(2,8,10,11)(4,12,9,16)(5,13,6,14)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,35)(34,38)(36,43)(37,41)(39,46)(40,48)(42,44)(45,47);(1,4)(2,5)(3,12)(6,10)(7,9)(8,13)(11,14)(15,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(34,39)(35,41)(36,44)(45,48);(1,5,7,6)(2,4,10,9)(3,13,15,14)(8,12,11,16)(18,19)(20,27)(24,26)(28,32)(33,36,37,44)(34,40,39,47)(35,42,41,43)(38,45,46,48);(1,5,7,6)(2,9,10,4)(3,14,15,13)(8,12,11,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(34,39)(35,41)(40,47)(42,43);(1,6,7,5)(2,4,10,9)(3,14,15,13)(8,12,11,16)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=gh,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012815);