Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| 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 | #35252 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2511 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)(7,9)(8,10)(11,12)(13,14,19,23,18,15,17,21)(16,22,26,27,24,20,25,28);(5,7)(6,9)(8,11)(10,12)(13,15,18,14)(16,20,24,22)(17,21,19,23)(25,28,26,27);(1,2,3,4)(6,10)(7,11)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28);(7,11)(9,12)(14,15)(20,22)(21,23)(27,28);(16,24)(20,22)(25,26)(27,28)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gh,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72308276, #72308296, #72308349 |
| maximal normal subgroup of | #72308276, #72308296, #72308349 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366766);