Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1319 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)(10,14)(11,12)(17,18,23,28)(19,30,33,24)(20,34,38,25)(21,29,39,26)(22,40,32,27)(31,41,48,46)(35,42,37,44)(36,43,47,45);(9,10)(11,15)(12,16)(13,14)(17,19,23,33)(18,24,28,30)(20,35,38,37)(21,32,39,22)(25,42,34,44)(26,27,29,40)(31,47,48,36)(41,43,46,45);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,21,36,23,38,39,47)(18,25,29,45,28,34,26,43)(19,31,22,35,33,48,32,37)(24,41,27,42,30,46,40,44);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,23,39)(18,26,28,29)(19,32,33,22)(20,36,38,47)(24,27,30,40)(25,43,34,45)(31,37,48,35)(41,42,46,44);(10,14)(15,16)(17,22,39,19,23,32,21,33)(18,27,26,24,28,40,29,30)(20,37,47,48,38,35,36,31)(25,44,43,46,34,42,45,41)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fhi,d^2=g,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6729710);