Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135848 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1853 |
| 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 | |
498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,19)(8,22,27,20)(9,16)(10,30)(11,17)(12,21)(13,18)(14,25,32,36)(15,29)(23,26,35,28)(24,31,33,34);(1,2)(3,4)(5,7,10,21,12,15,9,6,13,16,29,17,11,30,19,18)(8,23,25,14,28,22,24,31,27,35,36,32,26,20,33,34);(2,4)(5,8,9,24,11,26,10,25,13,27,19,33,12,28,29,36)(6,14,15,23,21,34,7,20,18,32,30,35,17,31,16,22)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=di,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250001, #72250007, #72250370, … |
| maximal normal subgroup of | #72250001, #72250007, #72250370, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56192);