Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10,13,16,14,17,12,21)(11,18,20,23,19,24,22,15);(6,7)(9,11)(10,15)(12,20)(13,22)(14,19)(16,24)(17,23)(18,21);(1,2,3,4)(5,6,8,7)(11,19)(15,23)(18,24)(20,22);(5,7,8,6)(11,19)(15,23)(18,24)(20,22);(9,12,14,13)(10,16,17,21)(11,20,19,22)(15,24,23,18)Polycyclic presentation:
2^9|a^2=f,c^2=ghi,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72315236, #72315247 |
| maximal normal subgroup of | #72315236, #72315247 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372747);