Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3412 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)(6,7)(9,10)(11,15)(12,22)(13,16)(14,17)(18,23)(19,20)(21,24);(5,6,8,7)(9,11)(10,15)(12,20)(13,23)(14,21)(16,18)(17,24)(19,22);(2,4)(5,7,8,6)(9,10)(11,18)(12,16)(13,22)(14,17)(15,23)(19,21)(20,24);(9,10)(11,19)(12,16)(13,22)(14,17)(15,20)(18,21)(23,24);(9,12,14,13)(10,16,17,22)(11,20,21,23)(15,18,24,19)Polycyclic presentation:
2^9|b^2=f,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307385, #72307466, #72307680 |
| maximal normal subgroup of | #72307385, #72307466, #72307680 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345858);