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 | #52164 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2015 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(7,11)(13,14,18,23,17,22,20,26)(15,19,25,28,24,16,21,27);(5,6)(7,10)(8,9)(11,12)(13,15,18,25,17,24,20,21)(14,19,23,28,22,16,26,27);(1,2,3,4)(13,14,20,26,17,22,18,23)(15,16,21,28,24,19,25,27);(6,9)(10,12)(13,16)(14,21)(15,23)(17,19)(18,27)(20,28)(22,25)(24,26);(5,7,8,11)(6,10,9,12)(14,22)(16,19)(23,26)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ci,d^b=dfi,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, 6303787);