Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C4×#102 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,9,16,12,18,11,28)(7,19,22,31,25,34,24,29)(8,14,26,33,27,17,10,15)(13,23,30,20,32,36,35,21);(5,7,10,23,11,24,27,21,12,25,26,36,9,22,8,20)(6,13,17,31,16,30,15,34,18,32,14,29,28,35,33,19);(1,2,3,4)(5,8,9,26,12,27,11,10)(6,14,16,33,18,17,28,15)(7,20,22,36,25,21,24,23)(13,29,30,19,32,31,35,34)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72314220, #72314225, #72314234, … |
| maximal normal subgroup of | #72314220, #72314225, #72314234, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54599);