Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45245 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
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)(19,24)(20,32)(21,26)(22,27)(23,28)(25,30)(29,38)(31,39)(33,37)(34,41)(35,42)(36,43)(40,44)(45,48)(46,47);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,33)(21,34)(22,30)(23,31)(25,27)(26,41)(28,39)(29,35)(32,37)(36,44)(38,42)(40,43)(45,46)(47,48);(10,14)(11,15)(17,20,23,30)(18,25,28,32)(19,22,31,33)(21,34,36,44)(24,37,39,27)(26,40,43,41)(29,35,45,46)(38,47,48,42);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,33,46,23,36,22,35)(18,26,37,47,28,43,27,42)(19,29,30,44,31,45,20,34)(24,38,25,40,39,48,32,41);(2,5)(6,8)(17,22,23,33)(18,27,28,37)(19,30,31,20)(21,35,36,46)(24,25,39,32)(26,42,43,47)(29,44,45,34)(38,40,48,41)Polycyclic presentation:
2^9|c^2=i,d^2=fhi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7516720);