Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #402001 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14828 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,6,10,19,8)(3,12,26,20,13,7,15,22)(4,16,17,23,18,9,24,25)(14,21,31,32,28,27,29,30);(1,3,5,15,6,13,19,26)(2,7,11,20,10,12,8,22)(4,14,17,29,18,28,24,31)(9,21,25,30,16,27,23,32);(2,8)(3,13)(5,19)(7,20)(9,23)(10,11)(12,22)(14,28)(16,25)(17,24)(21,30)(27,32);(1,4)(2,9,10,16)(3,14)(5,17)(6,18)(7,21,12,27)(8,23,11,25)(13,28)(15,29)(19,24)(20,30,22,32)(26,31)Polycyclic presentation:
2^9|a^2=fg,b^2=f,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72319788 |
| maximal normal subgroup of | #72319788 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309197);