Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 | |
1459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,6,4,8)(17,18,23,28)(19,24,32,41)(20,26,35,43)(21,27,22,37)(25,36,44,34)(29,39,33,45)(30,40,31,48)(38,47,42,46);(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,32)(18,24,28,41)(20,33,35,29)(21,30,22,31)(25,42,44,38)(26,45,43,39)(27,40,37,48)(34,47,36,46);(9,11)(10,13)(12,16)(14,15)(17,20,23,35)(18,25,28,44)(19,29,32,33)(21,34,22,36)(24,38,41,42)(26,27,43,37)(30,46,31,47)(39,40,45,48);(1,3,5,4)(2,6,7,8)(10,14)(11,16)(17,20)(18,26)(19,29)(21,36)(22,34)(23,35)(24,39)(25,37)(27,44)(28,43)(30,47)(31,46)(32,33)(38,48)(40,42)(41,45);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,27,28,37)(19,30,32,31)(20,34,35,36)(24,40,41,48)(25,43,44,26)(29,46,33,47)(38,45,42,39)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,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, 7214829);