Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,28)(22,26)(23,25)(24,27);(6,9)(10,12)(13,15)(14,18)(16,22)(17,21)(19,26)(20,28)(23,24)(25,27);(2,4)(14,19)(15,21)(17,24)(18,25)(22,23)(26,28);(5,7,8,11)(6,10,9,12);(15,22)(18,26)(21,23)(25,28)Polycyclic presentation:
2^9|a^2=gi,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ei,h^b=hi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72285202 |
| maximal normal subgroup of | #72285202 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339482);