Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
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×C22⋊C8 |
| prime radical | C2×#525 | C4 |
| LCS | G ▷ C2×C4 ▷ 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)(5,6,9,15,10,16,24,33,11,13,25,30,8,14,18,27)(7,17,20,34,19,28,35,32,21,31,23,36,22,12,26,29);(5,7,18,35,8,22,25,20,11,21,24,26,10,19,9,23)(6,12,27,34,14,31,30,29,13,28,33,36,16,17,15,32);(2,4);(6,13)(12,28)(14,16)(15,30)(17,31)(27,33)(29,32)(34,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^b=ei,g^b=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239418, #72239420, #72239433, … |
| maximal normal subgroup of | #72239418, #72239420, #72239433, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401603);