Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14366 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,20)(3,12,15,25,14,8,21,26)(4,16,18,9,19,28,22,27)(13,29,31,23,17,32,30,24);(1,3,5,21,7,14,6,15)(2,8,10,26,11,12,20,25)(4,17,18,30,19,13,22,31)(9,23,16,29,27,24,28,32);(1,4)(2,9)(3,13)(5,22)(6,18)(7,19)(8,23)(10,28)(11,27)(12,24)(14,17)(15,31)(16,20)(21,30)(25,29)(26,32);(1,4)(2,9)(3,13)(5,22)(6,18)(7,19)(8,24)(10,28)(11,27)(12,23)(14,17)(15,31)(16,20)(21,30)(25,32)(26,29)Polycyclic presentation:
2^9|a^2=h,b^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72237047, #72237064, #72237076, … |
| maximal normal subgroup of | #72237047, #72237064, #72237076, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291100);