Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509065 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,9)(8,10)(11,12)(13,14,20,23,22,24,28,26,17,15,18,21,19,16,25,27);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(14,15)(16,24)(21,23)(26,27);(2,4)(5,8,9,12)(6,10,7,11)(13,15,17,14)(16,22,24,19)(18,27,20,26)(21,28,23,25);(6,7)(10,11)(13,16)(14,19)(15,22)(17,24)(18,23)(20,21)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=fi,b^2=ei,c^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72343459, #72343833, #72343842 |
| maximal normal subgroup of | #72343459, #72343833, #72343842 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417149);