Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,23)(21,22);(1,2)(3,4)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,20,21)(18,22,23,24);(2,4)(5,6,8,7)(10,14)(13,15)(18,23)(22,24);(5,7,8,6)(11,16)(13,15)(18,22)(19,21)(23,24)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=ef,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72265650, #72265659, #72265680, … |
| maximal normal subgroup of | #72265650, #72265659, #72265680, … |
| maximal quotient of | #72322286 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 338616);