Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17386 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
5354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,14,18,21)(12,15)(13,16)(17,20,23,24)(19,22);(2,4)(9,11)(10,14,16,21)(12,17)(13,18)(15,20,22,24)(19,23);(6,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24);(12,19)(15,22)(17,23)(20,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,b^a=be,c^a=cf,e^a=ei,h^a=hi,e^b=ei,g^b=gi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72236375, #72237663, #72238859, … |
| maximal normal subgroup of | #72236375, #72237663, #72238859, … |
| maximal quotient of | #72234290, #72236441, #72238079, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260604);