Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13389 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2033 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
456 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)(5,6,12,18,10,22,29,16,14,19,11,17,13,7,9,21)(8,20,26,23,25,15,31,33,28,36,30,32,27,34,24,35);(5,7,10,19,14,22,13,6)(8,23,25,35,28,32,27,33)(9,21,12,17,29,16,11,18)(15,26,20,31,34,30,36,24);(5,8,9,24,13,27,11,30,14,28,29,31,10,25,12,26)(6,15,16,32,7,20,18,33,19,34,21,23,22,36,17,35)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=dh,f^b=fh,g^b=gh,h^b=hi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321505, #72321585, #72321598, … |
| maximal normal subgroup of | #72321505, #72321585, #72321598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56258);