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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2482 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)(15,22)(16,20)(17,25)(18,21)(19,24)(23,28)(26,27);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(15,23)(19,27)(22,28)(24,26);(2,4)(13,15,18,23)(14,19,21,27)(16,24,17,26)(20,28,25,22);(5,7,8,11)(6,10,9,12)(13,14)(15,19)(16,25)(17,20)(18,21)(22,24)(23,27)(26,28);(2,4)(6,9)(10,12)(13,16,18,17)(14,20,21,25)(15,24,23,26)(19,28,27,22)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72282949, #72282998, #72283332, … |
| maximal normal subgroup of | #72282949, #72282998, #72283332, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579021);