Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 |
| prime radical | #39362 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(7,11)(13,14)(15,19)(16,23)(17,18)(20,22)(21,26)(24,27)(25,28);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15,18,22)(14,20,17,19)(16,23,25,28)(21,26,24,27);(2,4)(13,16,18,25)(14,21,17,24)(15,23,22,28)(19,27,20,26);(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,20,22,19)(16,24,25,21)(23,26,28,27);(2,4)(6,9)(10,12)(14,17)(15,22)(21,24)(23,28)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72284213 |
| maximal normal subgroup of | #72284213 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579148);