Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #368884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14023 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60426 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1685 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,25)(3,14,17,9,18,29,26,19)(4,10,20,31,21,30,28,16)(5,22,23,11,24,15,27,32);(1,3,7,17,8,18,6,26)(2,9,12,29,13,19,25,14)(4,15,20,32,21,22,28,11)(5,16,23,10,24,31,27,30);(1,4)(2,10,13,30)(3,15,18,22)(5,19)(6,20)(7,28)(8,21)(9,24)(11,26,32,17)(12,16,25,31)(14,27)(23,29);(1,5,8,24)(2,11,13,32)(3,16,18,31)(4,19,21,9)(6,27,7,23)(10,26,30,17)(12,15,25,22)(14,28,29,20)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72339817 |
| maximal normal subgroup of | #72339817 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276083);