Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #348098 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #32926 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1182 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 188 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:
(2,5)(3,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,20)(19,21)(22,24);(1,2,3,5,4,6,7,8)(10,14)(15,20)(16,21)(22,23);(9,11)(10,15)(12,17)(13,18)(14,20)(16,22)(19,24)(21,23);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,24)(20,23)Polycyclic presentation:
2^9|b^2=ei,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^c=gi,f^d=fi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72268302, #72268303, #72268313, … |
| maximal normal subgroup of | #72268302, #72268303, #72268313, … |
| maximal quotient of | #72313245, #72342989, #72342994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 255300);