Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #249391 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
2124 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 2124 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,14)(12,15)(13,17)(16,19)(18,22)(20,23)(21,24);(3,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(2,5)(3,8)(6,7)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24);(9,11)(10,16)(12,18)(13,19)(14,17)(15,22)(20,24)(21,23)Polycyclic presentation:
2^9|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 | #72233804, #72247578, #72282958, … |
| maximal normal subgroup of | #72233804, #72247578, #72282958, … |
| maximal quotient of | #72283933 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 156587);