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 | #55101 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4083 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,4)(3,10,12,8)(5,14,6,9)(11,15,13,16)(17,18,21,24)(19,25,27,22)(20,28,30,23)(26,31,29,32)(33,34,37,40)(35,38,42,46)(36,39,44,48)(41,45,43,47);(1,3)(2,8)(4,10)(5,11)(6,13)(7,12)(9,15)(14,16)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,35,37,42)(34,38,40,46)(36,43,44,41)(39,47,48,45);(1,4,7,2)(3,10,12,8)(5,9,6,14)(11,15,13,16)(17,19,21,27)(18,22,24,25)(20,29,30,26)(23,32,28,31)(33,36)(34,39)(35,41)(37,44)(38,45)(40,48)(42,43)(46,47);(1,5,7,6)(2,9,4,14)(3,11,12,13)(8,15,10,16)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32)(36,44)(39,48)(41,43)(45,47);(3,12)(8,10)(11,13)(15,16)(18,24)(22,25)(23,28)(31,32)(35,42)(38,46)(41,43)(45,47);(1,6,7,5)(2,9,4,14)(3,13,12,11)(8,15,10,16)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=g,d^2=g,f^2=gh,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, 8009177);