Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213744 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #35346 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2494 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 2494 conjugacy classes of subgroups — too many to draw. Only its 234 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 — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14,16,17)(12,20,21,18)(13,19)(15,23)(22,24);(2,6)(5,8)(18,20)(22,23);(3,7)(5,8)(9,12)(10,15)(11,18)(13,21)(14,22)(16,24)(17,23)(19,20)Polycyclic presentation:
2^9|b^2=h,b^a=be,c^a=cf,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72210024, #72210026, #72225658, … |
| maximal normal subgroup of | #72210024, #72210026, #72225658, … |
| maximal quotient of | #72211163, #72235157, #72235262, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298905);