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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,22)(16,19)(17,28)(18,21)(20,26)(23,24)(25,27);(2,3)(6,9)(10,12)(13,15,18,26)(14,20,21,22)(16,27,28,23)(17,25,19,24);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16)(14,17)(15,23)(18,28)(19,21)(20,24)(22,25)(26,27);(1,3,4,2)(14,21)(15,24)(17,19)(20,23)(22,27)(25,26);(6,9)(10,12)(13,17,18,19)(14,16,21,28)(15,25,26,24)(20,27,22,23)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7467470);