Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457579 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2840 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1340 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 1340 conjugacy classes of subgroups — too many to draw. Only its 162 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:
(2,5)(6,8)(9,10)(11,17,19,14)(12,20,22,15)(13,16)(18,24)(21,23);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,21)(13,19)(15,23)(16,17)(18,22)(20,24);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,18,19,21)(13,22)(14,23,17,24)(16,20);(3,7)(6,8)(15,20)(23,24)Polycyclic presentation:
2^9|a^2=fg,b^2=egh,c^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=de,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72317900, #72317909, #72318437, … |
| maximal normal subgroup of | #72317900, #72317909, #72318437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381379);