Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #348098 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×H(5,𝔽2) | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1060 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 1060 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,13,16)(11,14,18,21)(12,15,19,22)(17,23,24,20);(1,2,3,5,4,6,7,8);(9,11)(10,14)(12,17)(13,18)(15,20)(16,21)(19,24)(22,23);(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,24)(21,23)Polycyclic presentation:
2^9|a^2=i,b^2=e,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, #72268307, #72268320, … |
| maximal normal subgroup of | #72268302, #72268307, #72268320, … |
| maximal quotient of | #72313299 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 255297);