Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4821 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(5,6)(7,8);(2,4)(9,10)(11,19)(12,17)(13,22)(14,18)(15,20)(16,23)(21,24);(6,8)(9,11)(10,15)(12,20)(13,23)(14,21)(16,18)(17,24)(19,22);(9,11)(10,16)(12,20)(13,23)(14,21)(15,18)(17,19)(22,24);(9,12,14,13)(10,17,18,22)(11,20,21,23)(15,24,16,19)Polycyclic presentation:
2^9|a^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72313563, #72315360, #72315401, … |
| maximal normal subgroup of | #72313563, #72315360, #72315401, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344859);