Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2014 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)(5,6)(7,12)(8,10)(9,11);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,14,17,21)(15,18,22,27)(16,24,26,20)(19,25,28,23);(2,4)(13,15,17,22)(14,18,21,27)(16,23,26,25)(19,20,28,24);(2,4)(5,8,9,7)(6,10,11,12)(13,15)(14,19)(16,25)(17,22)(18,24)(20,27)(21,28)(23,26);(6,11)(10,12)(15,22)(18,27)(19,28)(23,25)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72257557, #72257569, #72263859, … |
| maximal normal subgroup of | #72257557, #72257569, #72263859, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7424636);