Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×#535 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,10,11,9,7,8,12)(14,18,21,24)(20,25,27,28);(5,7,10,12,9,6,8,11)(13,14,19,18,17,21,16,24)(15,20,26,25,22,27,23,28);(1,2,3,4)(5,8,9,10)(6,11,7,12)(13,15)(14,20)(16,23)(17,22)(18,25)(19,26)(21,27)(24,28);(15,22)(20,27)(23,26)(25,28)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,e^2=h,g^2=i,b^a=be,c^a=cf,c^b=ci,e^b=eh,g^b=gh,d^c=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111697);