Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30871 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2675 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14361 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,8,13,6,12)(3,14,16,4,17,19,9,20)(5,10,23,27,24,29,22,28)(15,30,21,32,31,26,25,18);(1,3)(2,4)(5,21)(6,9)(7,16)(8,17)(10,26)(11,19)(12,14)(13,20)(15,22)(18,27)(23,31)(24,25)(28,32)(29,30);(1,4)(2,9)(3,11)(5,18)(6,14)(7,19)(8,20)(10,25)(12,17)(13,16)(15,27)(21,29)(22,26)(23,30)(24,32)(28,31);(1,5,6,22,8,24,7,23)(2,10,11,27,13,29,12,28)(3,15,16,21,17,31,9,25)(4,18,14,26,20,32,19,30)Polycyclic presentation:
2^9|a^2=h,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213603, #72213889, #72213913, … |
| maximal normal subgroup of | #72213603, #72213889, #72213913, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290755);