Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4⋊C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10)(11,19,22,15)(12,16)(13,17)(14,18)(20,23,24,21);(5,6,7,8)(9,11)(10,15)(12,20)(13,23)(14,22)(16,21)(17,24)(18,19);(2,4)(9,12,14,17)(10,16,18,13)(11,20,22,24)(15,21,19,23);(9,13,14,16)(10,17,18,12)(11,21)(15,24)(19,20)(22,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=gi,b^a=be,c^a=cf,e^a=ei,d^b=de,g^b=gi,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#1740, #72310626, #72310660 |
| maximal normal subgroup of | D4×#1740, #72310626, #72310660 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263687);