Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #365053 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2363 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13990 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
859 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)(3,13,16,25,17,24,15,8)(4,18,20,9,22,27,21,28)(14,26,30,29,19,23,31,32);(1,3)(2,8,12,25)(4,19,22,14)(5,15)(6,16)(7,17)(9,26)(10,24,11,13)(18,32)(20,31,21,30)(23,28)(27,29);(1,4,5,20,7,22,6,21)(2,9,11,18,12,28,10,27)(3,14,15,30,17,19,16,31)(8,23,13,29,25,26,24,32);(2,10,12,11)(8,24,25,13)(9,27,28,18)(23,32,26,29)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=e,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319525 |
| maximal normal subgroup of | #72319525 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273468);