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 |
| prime radical | C2×#3267 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(15,22)(19,23)(24,25)(27,28);(6,9)(10,12)(13,14,17,21)(15,23,22,19)(16,18,26,20)(24,27,25,28);(2,4)(5,7,8,11)(6,10,9,12)(14,18)(15,24)(16,26)(19,23)(20,21)(22,25);(13,15)(14,19)(16,25)(17,22)(18,27)(20,28)(21,23)(24,26);(14,20)(15,25)(16,26)(18,21)(19,23)(22,24)Polycyclic presentation:
2^9|b^2=i,c^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,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 | #72250121, #72256952, #72256983 |
| maximal normal subgroup of | #72250121, #72256952, #72256983 |
| maximal quotient of | #72264924, #72265036, #72265198, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482204);