Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #162342 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35031 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1529 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,10)(7,8)(11,12)(13,14)(15,18,22,25)(16,19)(17,20)(21,24,27,28)(23,26);(5,6,8,10,9,12,7,11)(13,15)(14,18,20,25)(16,21)(17,22)(19,24,26,28)(23,27);(2,4)(5,7,9,8)(6,11,12,10)(13,16,17,23)(14,19,20,26)(15,21,22,27)(18,24,25,28);(6,12)(10,11)(15,22)(16,23)(18,25)(19,26)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,e^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,g^b=gi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103649);