Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4414082 |
| automorphism | Aut(G) order 25165824 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5022 conjugacy classes of subgroups; only the 666 normal subgroups are shown.
This group has 5022 conjugacy classes of subgroups — too many to draw. Only its 666 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(11,15)(14,16)(18,21)(22,23);(10,13)(14,16)(17,18,20,21)(19,22,24,23);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,21)(22,24);(1,2)(3,4)(5,6)(7,8);(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,24)(21,23)Polycyclic presentation:
2^9|b^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=ef80 conjugacy classes — view the full character table.
| Fitting subgroup of | #72258928 |
| maximal subgroup of | #72258893, #72258901, #72258928 |
| maximal normal subgroup of | #72258893, #72258901, #72258928 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4321704);