Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #365053 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2363 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13984 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1145 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,10)(3,13,17,25,16,24,15,8)(4,18,20,28,21,9,22,27)(14,30,31,26,19,29,32,23);(1,3)(2,8,12,25)(4,19,21,14)(5,15)(6,17)(7,16)(9,26)(10,24,11,13)(18,23)(20,32,22,31)(27,29)(28,30);(1,4)(2,9)(3,14)(5,20)(6,22)(7,21)(8,23)(10,27)(11,28)(12,18)(13,29)(15,31)(16,19)(17,32)(24,30)(25,26);(2,10,12,11)(8,24,25,13)(9,27,18,28)(23,30,26,29)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72338601, #72338602, #72338609 |
| maximal normal subgroup of | #72338601, #72338602, #72338609 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273194);