Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,8)(10,11)(13,14,16,20,18,21,17,22)(15,23,24,28,25,27,26,19);(2,4)(9,10,12,11)(13,15,18,25)(14,19,21,28)(16,26,17,24)(20,27,22,23);(2,4)(14,20)(15,24)(16,17)(21,22)(23,27)(25,26);(6,8)(9,11,12,10)(14,21)(15,25)(20,22)(24,26)Polycyclic presentation:
2^9|a^2=fi,b^2=efi,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72311962, #72311971, #72311980, … |
| maximal normal subgroup of | #72311962, #72311971, #72311980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308179);