Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,33)(25,37)(30,39)(31,40)(32,41)(35,42)(36,44)(38,43)(45,46)(47,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,31,23,32,21,33)(18,24,26,41,28,40,27,29)(20,35,37,47,38,30,36,46)(25,42,34,45,44,39,43,48);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20,21,36,23,38,22,37)(18,25,27,43,28,44,26,34)(19,30,33,47,32,35,31,46)(24,39,29,45,40,42,41,48);(9,11)(10,13)(12,15)(14,16)(17,21,23,22)(18,26,28,27)(19,31,32,33)(20,36,38,37)(24,29,40,41)(25,34,44,43)(30,46,35,47)(39,45,42,48);(11,15)(13,16)(19,32)(24,40)(29,41)(30,35)(31,33)(39,42)(45,48)(46,47)Polycyclic presentation:
2^9|b^2=ghi,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=ehi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6512987);