Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141926 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1993 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,9,16,11,14,10,26)(7,17,18,29,21,12,20,30)(8,22,24,32,25,13,23,15)(19,31,33,27,34,36,35,28);(1,2,4,3)(5,7,10,18,11,21,9,20)(6,12,26,30,14,17,16,29)(8,19,23,33,25,34,24,35)(13,31,32,27,22,36,15,28);(1,3,4,2)(6,13,14,22)(7,18)(8,23)(9,10)(12,27,17,28)(15,16,32,26)(19,34)(20,21)(24,25)(29,36,30,31)Polycyclic presentation:
2^9|a^2=g,b^2=dh,c^2=d,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,g^b=gi,e^c=eh,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249871, #72249873, #72249885, … |
| maximal normal subgroup of | #72249871, #72249873, #72249885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55938);