Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #249391 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1701 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1701 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(12,19)(14,20)(15,22)(21,24);(3,8)(5,6)(9,10,13,16)(11,14,18,20)(12,15,19,22)(17,21,23,24);(2,5)(3,8)(6,7)(9,11)(10,14)(12,17)(13,18)(15,21)(16,20)(19,23)(22,24);(9,12)(10,15)(11,17)(13,19)(14,21)(16,22)(18,23)(20,24)Polycyclic presentation:
2^9|b^2=h,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,e^c=eh,f^c=fi,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72285006 |
| maximal normal subgroup of | #72285006 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 156588);