Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418769 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2804 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
747 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 747 conjugacy classes of subgroups — too many to draw. Only its 178 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:
(1,2,5,4)(3,7,8,6)(9,10)(11,21,17,12)(13,24,22,23)(14,19)(15,20)(16,18);(1,3,5,8)(2,6,4,7)(9,11,15,13,16,17,14,22)(10,12,20,23,18,21,19,24);(1,4,5,2)(3,6,8,7)(9,12)(10,17)(11,18)(13,19)(14,24)(15,23)(16,21)(20,22);(3,8)(6,7)(9,13,14,11,16,22,15,17)(10,12,19,24,18,21,20,23)Polycyclic presentation:
2^9|a^2=ef,b^2=efh,c^2=gh,d^2=efg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72227423 |
| maximal normal subgroup of | #72227423 |
| maximal quotient of | #72268633 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361143);