Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13391 |
| 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 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6,8,16,10,17,22,12,11,15,9,14,23,30,24,26)(7,13,18,27,21,28,33,35,20,29,19,31,32,36,34,25);(6,12,15,26)(7,18,20,19)(8,22)(9,24)(10,23)(14,17,16,30)(21,34,32,33)(25,31)(27,35)(28,36);(5,7,8,19,10,21,22,34,11,20,9,18,23,32,24,33)(6,13,14,27,17,28,26,35,15,29,16,31,30,36,12,25)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=h,f^2=h,g^2=h,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,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250059, #72250079, #72250104, … |
| maximal normal subgroup of | #72250059, #72250079, #72250104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56261);