Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13459 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #498 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2058 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
708 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)(7,17,20,34)(8,21)(9,13)(10,15)(11,16)(12,18,28,35)(14,25)(19,33,26,30)(22,27,36,29)(23,31)(24,32);(5,7)(6,12,16,28)(8,22)(9,19)(10,26)(11,20)(13,29,15,27)(14,30,32,33)(17,31,34,21)(18,25)(23,36)(24,35);(2,4)(6,13,16,15)(8,23)(12,27,28,29)(14,31,32,21)(17,33,34,30)(18,35)(22,36)(24,25)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=h,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=dh,g^b=gi,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72231508, #72231541, #72231555, … |
| maximal normal subgroup of | #72231508, #72231541, #72231555, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56649);