Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,10)(9,11);(5,7,9,8)(6,10,11,12)(13,14)(15,22)(16,23)(17,27)(18,21)(19,26)(20,28)(24,25);(1,2,3,4)(13,15,18,24)(14,16,21,26)(17,19,28,23)(20,25,27,22);(5,8,9,7)(6,10,11,12)(13,16,18,26)(14,19,21,23)(15,20,24,27)(17,25,28,22);(14,20)(16,25)(17,28)(19,23)(21,27)(22,26)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72256890, #72256912, #72257071 |
| maximal normal subgroup of | #72256890, #72256912, #72257071 |
| maximal quotient of | #72264237, #72265134 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6491652);