Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,28,22,27,37,26)(19,29,32,41,33,23,38,42)(20,25,36,44,35,45,34,24)(30,40,48,39,31,46,47,43);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,31)(21,32)(22,33)(24,43)(25,40)(26,41)(27,29)(28,42)(30,35)(34,48)(36,47)(37,38)(39,44)(45,46);(9,11)(10,13)(12,16)(14,15)(17,20,22,35)(18,24,27,44)(19,30,33,31)(21,34,37,36)(23,39,29,43)(25,28,45,26)(32,47,38,48)(40,41,46,42);(3,8)(6,7)(11,16)(13,15)(17,20)(18,25)(19,31)(21,36)(22,35)(23,40)(24,26)(27,45)(28,44)(29,46)(30,33)(32,47)(34,37)(38,48)(39,42)(41,43);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,37)(18,26,27,28)(19,32,33,38)(20,34,35,36)(23,41,29,42)(24,25,44,45)(30,47,31,48)(39,46,43,40)Polycyclic presentation:
2^9|a^2=fgi,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6918279);