Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | #34829 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,9,12)(7,11,8,10)(13,14,19,23,18,20,16,22)(15,24,25,28,26,21,27,17);(5,6)(7,10)(8,11)(9,12)(14,20)(17,28)(21,24)(22,23);(2,4)(5,7,9,8)(6,10,12,11)(13,15,18,26)(14,21,20,24)(16,27,19,25)(17,22,28,23);(5,7,9,8)(6,11,12,10)(13,16,18,19)(14,22,20,23)(15,25,26,27)(17,24,28,21);(5,8,9,7)(6,11,12,10)(13,17,19,24,18,28,16,21)(14,15,23,25,20,26,22,27)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=efi,d^b=dfi,e^b=ei,d^c=di,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6307551);