Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,8)(9,10,11,12)(13,14,19,20)(15,18,17,16);(1,2)(3,4)(13,15,18,20,19,17,16,14);(6,8)(13,16,19,18)(14,15,20,17);(2,4)(13,17)(14,16)(15,19)(18,20)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,c^b=ci,d^b=de,e^d=ei,h^d=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72264158, #72264161, #72264237, … |
| maximal normal subgroup of | #72264158, #72264161, #72264237, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263291);