Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2089 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)(5,6)(7,10)(8,9)(11,12)(13,14,17,21,16,20,24,28)(15,19,23,27,22,18,25,26);(1,2)(3,4)(13,15)(14,18)(16,22)(17,23)(19,20)(21,26)(24,25)(27,28);(6,9)(10,12)(13,15)(14,19)(16,22)(17,23)(18,20)(21,27)(24,25)(26,28);(2,4)(14,20)(15,22)(21,28)(23,25);(5,7,8,11)(6,10,9,12)(15,22)(18,19)(23,25)(26,27)Polycyclic presentation:
2^9|a^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=di,d^b=df,e^b=ei,d^c=di,e^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288226);