Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394481 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2472 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14817 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1057 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)(3,13,16,22)(4,17)(5,10)(6,11)(7,12)(8,14,24,20)(9,18)(15,29,30,23)(19,25)(21,26)(27,28,32,31);(3,14,16,20)(8,22,24,13)(15,28,30,31)(23,32,29,27);(1,3,5,20,7,16,6,14)(2,8,10,22,12,24,11,13)(4,15,18,31,19,30,21,28)(9,23,25,32,26,29,17,27);(1,4,5,18,7,19,6,21)(2,9,10,25,12,26,11,17)(3,15,20,31,16,30,14,28)(8,23,22,32,24,29,13,27)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^c=ei,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319143, #72319406, #72319438, … |
| maximal normal subgroup of | #72319143, #72319406, #72319438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308670);