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 | #52096 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2229 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)(13,14,20,24,19,18,17,23)(15,25,26,22,27,21,16,28);(6,9)(7,11)(13,15)(14,21)(16,17)(18,25)(19,27)(20,26)(22,23)(24,28);(1,2,3,4)(13,16,19,26)(14,22,18,28)(15,17,27,20)(21,23,25,24);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,23,18,24)(15,26,27,16)(21,28,25,22);(6,9)(10,12)(13,18,20,23,19,14,17,24)(15,21,26,28,27,25,16,22)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=efi,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, 6311412);