Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,8)(9,10,11,12)(13,14,18,22,19,24,17,23)(15,20,25,21,16,27,26,28);(1,2,4,3)(5,6)(7,8)(10,12)(13,15,18,25,19,16,17,26)(14,20,22,21,24,27,23,28);(6,8)(10,12)(13,16,19,15)(14,21,24,28)(17,26,18,25)(20,23,27,22);(1,3,4,2)(10,12)(13,17,19,18)(14,22,24,23)(15,25,16,26)(20,28,27,21)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=i,d^2=efgi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279875);