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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1691 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)(4,6)(5,8)(17,18)(19,28)(20,25)(21,26)(22,37)(23,27)(24,32)(29,44)(30,45)(31,38)(33,40)(34,48)(35,42)(36,43)(39,46)(41,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(19,29)(20,33)(24,38)(25,40)(28,44)(31,32)(35,36)(42,43);(9,11)(10,13)(12,16)(14,15)(17,19,23,29)(18,24,27,38)(20,34,33,30)(21,32,22,31)(25,41,40,39)(26,44,37,28)(35,46,36,47)(42,48,43,45);(3,8)(5,7)(10,14)(11,16)(17,20,23,33)(18,25,27,40)(19,30,29,34)(21,36,22,35)(24,39,38,41)(26,43,37,42)(28,45,44,48)(31,47,32,46);(2,6)(5,7)(10,14)(13,15)(17,21,23,22)(18,26,27,37)(19,31,29,32)(20,35,33,36)(24,28,38,44)(25,42,40,43)(30,46,34,47)(39,48,41,45)Polycyclic presentation:
2^9|c^2=i,d^2=i,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=ehi,d^c=dh,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, 7208311);