Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34727 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1989 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)(5,6)(7,10)(8,12)(9,11)(13,14,16,19,17,20,24,27)(15,21,23,28,22,18,25,26);(5,7,10,6)(8,12,9,11)(15,22)(18,21)(23,25)(26,28);(2,4)(5,8,10,9)(6,11,7,12)(15,22)(18,21)(23,25)(26,28);(5,8,10,9)(6,12,7,11);(5,9,10,8)(6,12,7,11)(13,15,16,23,17,22,24,25)(14,18,19,26,20,21,27,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,e^a=efi,d^b=df,e^b=ei,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72308891, #72308918, #72308960, … |
| maximal normal subgroup of | #72308891, #72308918, #72308960, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6308075);