Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196038 |
| 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 | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1378 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)(5,6,10,11,9,12,8,7)(13,14,18,21,17,20,26,28)(15,16,23,25,22,24,19,27);(5,7,10,12,9,11,8,6)(13,15,18,23,17,22,26,19)(14,16,21,25,20,24,28,27);(2,4)(5,8,9,10)(6,11,12,7)(13,16)(14,19)(15,21)(17,24)(18,25)(20,23)(22,28)(26,27);(5,8,9,10)(6,11,12,7)(14,20)(16,24)(21,28)(25,27)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=h,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=eh,g^b=gh,d^c=dhi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 103264);