Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137659 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1893 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,16,18,11)(8,20)(9,25)(10,13)(12,17,31,21)(14,23)(15,27)(19,29,26,34)(22,32,36,28)(24,30)(33,35);(1,2)(3,4)(5,7)(6,11)(8,21)(9,19)(10,18)(12,30)(13,16)(14,32)(15,29)(17,24)(20,31)(22,35)(23,36)(25,34)(26,27)(28,33);(2,4)(5,7)(6,12,13,31)(8,22)(9,26)(10,18)(11,20,16,30)(14,34,33,29)(15,28,25,32)(17,23)(19,27)(21,35)(24,36)Polycyclic presentation:
2^9|a^2=g,c^2=d,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,e^c=eh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321662, #72321701, #72321712, … |
| maximal normal subgroup of | #72321662, #72321701, #72321712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55801);