Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1601 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(18,23)(19,27)(20,31)(26,35)(30,42)(33,36)(38,43)(45,48);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,18,21,23)(19,28,27,24)(20,32,31,40)(22,35,34,26)(25,36,37,33)(29,45,41,48)(30,39,42,44)(38,46,43,47);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,19,21,27)(18,24,23,28)(20,29,31,41)(22,30,34,42)(25,38,37,43)(26,39,35,44)(32,48,40,45)(33,46,36,47);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,20)(18,25)(19,29)(21,31)(22,36)(23,37)(24,38)(26,40)(27,41)(28,43)(30,47)(32,35)(33,34)(39,45)(42,46)(44,48);(2,6)(7,8)(19,27)(20,31)(24,28)(25,37)(30,42)(32,40)(33,36)(39,44)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=f,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7433101);