Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324436 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1548 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,22,23)(19,27,30,42)(20,25,31,37)(21,34,35,26)(24,38,36,29)(28,43,44,48)(32,41,33,40)(39,45,47,46);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,23)(20,31)(24,36)(26,34)(27,42)(28,44)(32,33)(45,46);(2,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,29,22,30,35,38)(18,24,26,27,23,36,34,42)(20,28,33,46,31,44,32,45)(25,39,41,43,37,47,40,48);(3,8)(5,7)(10,14)(11,16)(17,20)(18,25)(19,28)(21,33)(22,31)(23,37)(24,39)(26,41)(27,43)(29,46)(30,44)(32,35)(34,40)(36,47)(38,45)(42,48);(2,6)(5,7)(10,14)(13,15)(17,21,22,35)(18,26,23,34)(19,29,30,38)(20,32,31,33)(24,27,36,42)(25,40,37,41)(28,45,44,46)(39,48,47,43)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=fgi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7234237);