Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,9)(10,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,25)(24,27)(26,28);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15)(14,17)(16,20)(18,19);(2,4)(5,7,8,11)(6,10,9,12)(14,18)(17,19);(6,9)(10,12)(15,20)(17,19)(22,25)(23,28)(26,27)Polycyclic presentation:
2^9|b^2=ef,c^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 | #72308295, #72308317, #72308378, … |
| maximal normal subgroup of | #72308295, #72308317, #72308378, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 333133);