Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1806 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)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,22,24)(19,27,30,32)(20,26,21,25)(23,29,31,28);(2,4)(17,19,22,30)(18,23,24,31)(20,29,21,28)(25,32,26,27);(6,8)(19,28)(20,21)(23,27)(25,26)(29,30)(31,32);(9,11,12,16)(10,13,14,15)(18,24)(23,31)(25,26)(27,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265020, #72265052, #72265079 |
| maximal normal subgroup of | #72265020, #72265052, #72265079 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321854);