Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35512 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3518 conjugacy classes of subgroups; only the 262 normal subgroups are shown.
This group has 3518 conjugacy classes of subgroups — too many to draw. Only its 262 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:
(1,2)(3,4)(5,6,7,8)(9,10)(11,19)(12,13)(14,16)(15,20)(17,18)(21,24)(22,23);(9,11,14,21)(10,15,16,23)(12,19,18,24)(13,22,17,20);(2,4)(10,16)(13,17)(19,24)(20,22);(6,8)(9,12)(10,17)(11,15)(13,16)(14,18)(19,22)(20,24)(21,23)Polycyclic presentation:
2^9|a^2=hi,b^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72227754, #72227757, #72227759, … |
| maximal normal subgroup of | #72227754, #72227757, #72227759, … |
| maximal quotient of | #72267306 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356679);