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 | #2675 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,23,18,16,21,26)(15,24,25,17,22,20,28,27);(6,9)(7,11)(13,15,19,25,18,22,21,28)(14,20,23,27,16,24,26,17);(1,2,3,4)(13,16,19,26,18,14,21,23)(15,20,25,27,22,24,28,17);(5,7,8,11)(6,10,9,12)(13,17,18,27)(14,22,16,15)(19,20,21,24)(23,28,26,25);(6,9)(10,12)(14,16)(17,27)(20,24)(23,26)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi,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, 6303568);