Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480482 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #35676 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,18,10)(8,12)(9,22)(11,14)(13,25)(16,29,32,20)(17,24,34,30)(19,28)(21,27)(23,26)(31,35,33,36);(5,7,15,14)(6,11,18,10)(8,16,29,27)(9,17,30,22)(12,21,32,20)(13,25,34,24)(19,31,36,28)(23,33,35,26);(2,4)(5,8,11,21)(6,12,14,27)(7,16,18,32)(9,23,25,19)(10,20,15,29)(13,28,22,26)(17,33,34,31)(24,36,30,35);(5,9,11,25)(6,13,14,22)(7,17,18,34)(8,19,21,23)(10,24,15,30)(12,26,27,28)(16,31,32,33)(20,35,29,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312597, #72312610, #72312632, … |
| maximal normal subgroup of | #72312597, #72312610, #72312632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401920);