Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #385905 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14417 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60454 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1428 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)(3,9,15,28)(4,18,20,26)(5,22)(6,13)(7,12)(8,25)(10,21,30,14)(11,24)(16,27,17,32)(19,31)(23,29);(1,3)(2,9)(4,19)(5,21)(6,15)(7,17)(8,16)(10,29)(11,26)(12,32)(13,28)(14,23)(18,31)(20,24)(22,30)(25,27);(1,4,7,14,6,20,8,21)(2,10,12,26,13,30,25,18)(3,5,16,24,15,23,17,19)(9,11,27,29,28,31,32,22);(1,5,6,23)(2,11,13,31)(3,14,15,21)(4,16,20,17)(7,24,8,19)(9,26,28,18)(10,27,30,32)(12,29,25,22)Polycyclic presentation:
2^9|a^2=h,c^2=eg,d^2=i,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,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72215958, #72216181, #72216217, … |
| maximal normal subgroup of | #72215958, #72216181, #72216217, … |
| maximal quotient of | C2×#385906, #72252175, #72286836 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293102);