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 | #34741 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1963 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)(6,9)(10,12)(13,14,19,23,18,22,25,24)(15,17,21,26,20,27,28,16);(5,6)(7,10)(8,9)(11,12)(13,15,19,21,18,20,25,28)(14,17,23,26,22,27,24,16);(2,4)(13,16)(14,20)(15,22)(17,19)(18,26)(21,24)(23,28)(25,27);(6,9)(10,12)(13,17)(14,21)(15,24)(16,25)(18,27)(19,26)(20,23)(22,28);(5,7,8,11)(6,10,9,12)(14,22)(16,26)(17,27)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,d^b=dfi,d^c=di,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289637);