Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C22⋊C4 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C22×C22⋊C8 |
| prime radical | C2×#525 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1368 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)(5,6,8,13,9,14,19,28,10,12,20,22,21,26,32,33)(7,11,16,24,17,25,30,35,15,23,18,27,29,34,31,36);(5,7,8,18,9,17,19,31,10,15,20,16,21,29,32,30)(6,11,22,24,14,25,33,35,12,23,13,27,26,34,28,36);(2,4);(6,12)(11,23)(13,22)(14,26)(24,27)(25,34)(28,33)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,e^b=ei,g^b=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239420, #72239422, #72239432, … |
| maximal normal subgroup of | #72239420, #72239422, #72239432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401620);