Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×C4⋊C8 |
| prime radical | C25 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,15,10,14,23,30,11,13,24,29,8,20,22,31)(7,16,18,32,17,12,25,27,19,28,33,36,21,26,34,35);(5,7,9,18,10,17,23,25,11,19,24,33,8,21,22,34)(6,12,15,27,14,28,30,36,13,26,29,35,20,16,31,32);(2,4);(6,13)(12,26)(14,20)(15,29)(16,28)(27,35)(30,31)(32,36)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,e^b=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239421, #72239422, #72239433, … |
| maximal normal subgroup of | #72239421, #72239422, #72239433, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401418);