Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2318 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)(16,24)(21,28)(23,26)(25,27);(5,6,8,9)(7,10,11,12)(13,14,18,22)(15,20,17,19)(16,25,24,27)(21,26,28,23);(2,4)(14,19)(15,17)(20,22)(21,25)(23,26)(27,28);(5,7,8,11)(6,10,9,12)(13,15,18,17)(14,20,22,19)(16,23,24,26)(21,27,28,25);(6,9)(10,12)(13,16,18,24)(14,21,22,28)(15,23,17,26)(19,25,20,27)Polycyclic presentation:
2^9|b^2=fi,d^2=fi,e^2=i,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 | #72308202, #72308717 |
| maximal normal subgroup of | #72308202, #72308717 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6578683);