Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,25)(21,33)(22,27)(24,32)(26,35)(28,37)(29,38)(30,45)(31,41)(34,40)(36,42)(39,44)(43,48)(46,47);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,29)(18,23,27,38)(20,28,31,43)(21,30,35,44)(24,36,40,47)(25,37,41,48)(26,39,33,45)(32,42,34,46);(11,15)(13,16)(17,20)(18,24)(19,28)(21,34)(22,31)(23,36)(25,33)(26,41)(27,40)(29,43)(30,46)(32,35)(37,45)(38,47)(39,48)(42,44);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20)(18,25)(19,28)(21,32)(22,31)(23,37)(24,33)(26,40)(27,41)(29,43)(30,42)(34,35)(36,45)(38,48)(39,47)(44,46);(2,5)(6,8)(11,15)(13,16)(19,29)(20,31)(23,38)(24,40)(25,41)(30,44)(32,34)(39,45)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,e^c=ei,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, 7109426);