Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13374 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2023 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
502 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)(7,18)(8,21,24,19)(9,15)(10,28)(11,20)(12,17)(13,23,30,33)(14,27)(16,26)(22,31,32,34)(25,29,36,35);(1,2)(3,4)(5,7,10,20,11,14,9,6,12,15,27,16,26,28,18,17)(8,21,23,31,25,35,22,30,24,19,33,34,36,29,32,13);(2,4)(5,8,9,22,26,36,10,23,12,24,18,32,11,25,27,33)(6,13,14,29,20,34,7,19,17,30,28,35,16,31,15,21)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=hi,e^2=h,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^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250355, #72250362, #72250382, … |
| maximal normal subgroup of | #72250355, #72250362, #72250382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56190);