Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13373 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #486 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #2099 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,9,18,12,17,11,15)(7,16,10,13,21,30,20,29)(8,22,23,35,25,28,27,36)(14,31,34,19,33,26,32,24);(1,2,4,3)(5,7,9,10,12,21,11,20)(6,13,15,16,17,29,18,30)(8,19,23,26,25,24,27,31)(14,22,32,36,33,28,34,35);(1,3,4,2)(5,8,9,23,12,25,11,27)(6,14,15,32,17,33,18,34)(7,19,10,26,21,24,20,31)(13,28,16,35,29,22,30,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=hi,g^2=i,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,e^c=ehi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250420, #72250433, #72250437, … |
| maximal normal subgroup of | #72250420, #72250433, #72250437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55924);