Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13460 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2058 |
| 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 | |
530 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)(5,6,12,15)(7,9,18,11)(8,19)(10,17,27,35)(13,28)(14,26,32,24)(16,33)(20,31)(21,30)(22,34)(23,36)(25,29);(1,2,4,3)(5,7,9,15,12,18,11,6)(8,13,20,16,23,30,22,29)(10,17,24,32,27,35,26,14)(19,33,34,28,36,25,31,21);(1,3,4,2)(5,8,9,20,12,23,11,22)(6,13,7,16,15,30,18,29)(10,25,24,21,27,33,26,28)(14,31,17,19,32,34,35,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,f^b=fi,e^c=eh,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249929, #72249941, #72249964, … |
| maximal normal subgroup of | #72249929, #72249941, #72249964, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55925);