Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115640 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#2431 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
2038 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 2038 conjugacy classes of subgroups — too many to draw. Only its 150 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 — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,17,14,18,12,21)(11,16,19,15,20,24,22,23);(9,11)(10,15)(12,22)(13,19)(14,20)(16,21)(17,24)(18,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,19)(14,20)(15,17)(18,24)(21,23)Polycyclic presentation:
2^9|a^2=g,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=dh,d^c=dh,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72233797, #72233805, #72233806, … |
| maximal normal subgroup of | #72233797, #72233805, #72233806, … |
| maximal quotient of | #72233898 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22844);