Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13442 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2054 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
730 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)(5,6,9,15,8,22,23,31,11,16,25,30,10,14,24,13)(7,17,20,27,19,12,26,28,18,32,33,36,21,29,34,35);(5,7,8,19,11,18,10,21)(6,12,14,17,16,29,22,32)(9,20,23,26,25,33,24,34)(13,27,30,35,31,36,15,28);(2,4)(6,13)(7,18)(8,10)(9,24)(12,27)(14,15)(16,31)(17,28)(20,26)(22,30)(23,25)(29,36)(32,35)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eh,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250420, #72250430, #72250442, … |
| maximal normal subgroup of | #72250420, #72250430, #72250442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55425);