Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| 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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(9,10)(11,15)(12,14)(13,16)(17,18,23,25,22,20,21,31)(19,26,30,32,29,24,28,27);(5,6,7,8)(9,11,12,16)(10,13,14,15)(17,19,21,28,22,29,23,30)(18,24,31,32,20,26,25,27);(2,4)(6,8)(17,20)(18,22)(19,27)(21,31)(23,25)(24,30)(26,28)(29,32);(10,14)(13,15)(17,21,22,23)(18,25,20,31)(19,28,29,30)(24,27,26,32)Polycyclic presentation:
2^9|a^2=h,b^2=eg,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fi,g^c=gi,h^c=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72264913, #72264943, #72264981, … |
| maximal normal subgroup of | #72264913, #72264943, #72264981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268946);