Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453948 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,9)(10,12)(13,14)(15,23)(16,17)(18,21)(19,24)(20,26)(22,28)(25,27);(2,3)(5,6,8,9)(7,10,11,12)(13,15,17,24)(14,19,16,23)(18,26,28,25)(20,21,27,22);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,20,24,27)(18,21,28,22)(19,26,23,25);(2,3)(7,11)(10,12)(13,15,17,24)(14,20,16,27)(18,25,28,26)(19,21,23,22)Polycyclic presentation:
2^9|b^2=efi,c^2=e,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363137);