Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,24,22,20);(1,2,3,4)(5,6)(7,8)(9,11,12,15)(10,13,14,16)(17,19,20,18,21,22,24,23);(6,8)(10,14)(13,16);(2,4)(18,19)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=i,b^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=dei,e^d=ei,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72264252, #72264326, #72264342, … |
| maximal normal subgroup of | #72264252, #72264326, #72264342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328230);