Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13417 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2041 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6,10,15,11,17,27,16)(7,18,13,26,20,34,14,9)(8,21,25,19,24,30,29,22)(12,28,32,35,31,36,33,23);(5,7,9,6,10,14,18,16,11,20,26,17,27,13,34,15)(8,22,23,31,25,30,28,32,24,19,35,12,29,21,36,33);(5,8,9,23,10,25,18,28,11,24,26,35,27,29,34,36)(6,12,13,30,16,33,7,19,17,31,14,21,15,32,20,22)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,d^b=di,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320262, #72320338, #72320467, … |
| maximal normal subgroup of | #72320262, #72320338, #72320467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56195);