Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | #23861 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,18)(17,25)(19,23)(20,24)(22,28)(26,27);(1,2)(3,4)(6,9)(10,12)(14,18)(19,27)(20,25)(21,28);(2,4)(13,15,16,22)(14,19,18,27)(17,26,24,23)(20,28,25,21);(2,4)(15,22)(19,27)(21,28)(23,26);(5,7,8,11)(6,10,9,12)(15,22)(19,27)(21,28)(23,26)Polycyclic presentation:
2^9|c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239589, #72239609, #72251240, … |
| maximal normal subgroup of | #72239589, #72239609, #72251240, … |
| maximal quotient of | #72239771, #72264418, #72264502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347305);