Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1221 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,23)(16,19)(17,21)(18,22)(20,24)(25,27)(26,28);(5,7,9,6)(8,11,12,10)(13,14)(15,20)(16,21)(17,19)(18,22)(23,24)(25,28)(26,27);(5,8,9,12)(6,10,7,11)(14,19)(16,17)(20,23)(21,22)(24,26)(27,28);(6,7)(10,11)(13,15)(14,20)(16,26)(17,24)(18,25)(19,23)(21,27)(22,28)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290615);