Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806527 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #39852 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1944 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,10)(7,8)(13,14)(15,20)(16,21)(17,19)(18,22)(23,26)(24,28)(25,27);(5,6)(7,12)(8,11)(9,10)(14,19)(15,16)(18,24)(20,27)(21,26)(23,25);(2,4)(5,7,9,8)(6,11,10,12)(13,15,18,16,17,23,24,25)(14,20,22,21,19,26,28,27);(5,8,9,7)(6,11,10,12)(14,19)(20,26)(21,27)(22,28);(13,16,24,15,17,25,18,23)(14,21,28,20,19,27,22,26)Polycyclic presentation:
2^9|c^2=h,d^2=fi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72308311, #72308331 |
| maximal normal subgroup of | #72308311, #72308331 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717791);