Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514060 |
| 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 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,8,9)(7,11,12,10)(15,22)(19,27)(23,28)(24,26);(2,3)(6,9)(10,11)(13,14,18,21)(15,23,22,28)(16,20,17,25)(19,26,27,24);(1,2,4,3);(1,3,4,2)(13,15,18,22)(14,19,21,27)(16,24,17,26)(20,28,25,23);(5,7,8,12)(6,10,9,11)(13,16,18,17)(14,20,21,25)(15,24,22,26)(19,28,27,23)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469949);