Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201103 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5,7,8)(4,6)(9,10)(11,19)(12,23,25,16)(13,26)(14,18)(15,21)(17,28)(20,33,36,40)(22,34)(24,31,27,32)(29,35,39,37)(30,38);(1,3)(2,5,6,8)(4,7)(9,11,13,21,14,22,28,38)(10,15,17,19,18,30,26,34)(12,20,27,37,25,36,24,35)(16,29,32,33,23,39,31,40);(9,12)(10,16)(11,20)(13,27)(14,25)(15,29)(17,32)(18,23)(19,33)(21,37)(22,36)(24,28)(26,31)(30,39)(34,40)(35,38);(2,6)(5,8)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108299);