Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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 | #24275 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2251 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)(19,24)(20,22)(23,27)(26,28);(1,2)(3,4)(6,9)(10,12)(14,18)(19,26)(20,25)(21,27);(2,4)(13,15,22,28,16,23,17,24)(14,19,25,27,18,26,20,21);(2,4)(15,23)(19,26)(21,27)(24,28);(5,7,8,11)(6,10,9,12)(15,23)(19,26)(21,27)(24,28)Polycyclic presentation:
2^9|c^2=gi,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=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72250065, #72256938, #72256979, … |
| maximal normal subgroup of | #72250065, #72256938, #72256979, … |
| maximal quotient of | #72264477, #72264522, #72264885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352272);