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 | #34736 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2951 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,15,17,20,25,22)(16,19,24,21,23,26,28,27);(5,7)(6,9)(8,12)(10,11)(13,15,18,20,17,22,25,14)(16,21,24,26,23,27,28,19);(2,4)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28);(7,12)(9,11);(6,10)(9,11)(14,20)(15,22)(16,23)(24,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,e^b=ei,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72282177, #72282202, #72282851 |
| maximal normal subgroup of | #72282177, #72282202, #72282851 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6294903);