Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29634 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,24)(23,26)(27,28);(6,9)(10,12)(13,15)(14,18)(16,21)(17,22)(19,25)(20,24)(23,27)(26,28);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(18,24)(25,28);(7,11)(10,12)(13,16,17,23)(14,19,20,26)(15,21,22,27)(18,25,24,28)Polycyclic presentation:
2^9|c^2=h,d^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,f^b=fi,d^c=de74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289584);