Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188176 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#547 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4⋊C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10)(11,19,22,15)(12,16)(13,23,20,17)(14,18)(21,24);(5,6)(7,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19);(2,4)(6,8)(9,12,14,21)(10,16,18,24)(11,20,22,13)(15,23,19,17);(9,13,14,20)(10,17,18,23)(11,21,22,12)(15,24,19,16)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72313290 |
| maximal normal subgroup of | #72313290 |
| maximal quotient of | #72313296, #72313341, #72313588, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266778);