Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188176 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#553 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4⋊C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(13,14)(15,21,24,18)(16,25,26,19)(17,20)(22,27)(23,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,23)(17,24)(19,28)(20,21)(22,26)(25,27);(6,9)(10,12)(13,16)(14,19)(15,22)(17,26)(18,27)(20,25)(21,28)(23,24);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,23)(17,26)(18,28)(20,25)(21,27)(22,24)Polycyclic presentation:
2^9|a^2=h,d^2=gi,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 270401);