Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2002 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,9)(10,11)(13,14)(15,17)(16,24)(18,21)(19,22)(20,25)(23,27)(26,28);(5,6,8,9)(7,11,12,10)(13,14)(15,18)(16,20)(17,21)(19,22)(23,26)(24,25)(27,28);(13,15,18,22,19,21,17,14)(16,23,26,28,27,24,25,20);(2,4)(5,7,8,12)(6,10,9,11)(13,16,17,25,19,27,18,26)(14,20,21,24,22,28,15,23)Polycyclic presentation:
2^9|b^2=e,c^2=fi,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 338115);