Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1862 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)(13,14,17,21)(15,22,18,19)(16,24,26,20)(23,27,25,28);(1,2,4,3)(5,6,8,9)(7,10,11,12)(13,15,17,18)(14,19,21,22)(16,25,26,23)(20,28,24,27);(1,3,4,2)(6,9)(7,11)(14,19)(15,18)(16,26)(20,28)(21,22)(24,27);(5,7)(6,10)(8,11)(9,12)(13,16,17,26)(14,20,21,24)(15,23,18,25)(19,27,22,28);(6,9)(7,11)(16,26)(20,24)(23,25)(27,28)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6505702);