Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2091 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)(6,9)(7,11)(13,14)(15,22)(16,18)(17,26)(19,21)(20,24)(23,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,20,16,22)(17,25,19,27)(21,28,26,23);(2,4)(13,16,18,14)(15,23,24,28)(17,26,19,21)(20,27,22,25);(5,7,8,11)(6,10,9,12)(14,16)(20,22)(21,26)(23,28);(6,9)(10,12)(13,17)(14,21)(15,24)(16,26)(18,19)(23,28)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72308367 |
| maximal normal subgroup of | #72308367 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6577390);