Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14363 |
| prime radical | #60415 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1343 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)(3,12,16,8)(4,17,20,27)(5,10)(6,21)(7,11)(9,19,26,22)(13,25)(14,28,15,24)(18,23)(29,31)(30,32);(1,3)(2,8)(4,18)(5,15)(6,14)(7,16)(9,25)(10,28)(11,12)(13,19)(17,29)(20,31)(21,24)(22,30)(23,27)(26,32);(1,4)(2,9,11,26)(3,13)(5,19)(6,22)(7,20)(8,23,12,29)(10,17,21,27)(14,18)(15,31)(16,30)(24,32,28,25);(2,10,11,21)(8,24,12,28)(9,17,26,27)(23,32,29,25)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318867, #72318876, #72318963, … |
| maximal normal subgroup of | #72318867, #72318876, #72318963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290932);