Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,14,10,12,16,24)(7,15,18,26,19,30,23,11)(8,13,21,29,22,27,31,35)(17,28,20,33,32,36,34,25);(5,7,16,18,10,19,9,23)(6,11,24,15,12,26,14,30)(8,20,31,32,22,34,21,17)(13,28,35,33,27,36,29,25);(6,12)(13,27)(14,24)(29,35);(2,4)(5,8,9,21,10,22,16,31)(6,13,14,29,12,27,24,35)(7,17,18,20,19,32,23,34)(11,25,15,28,26,33,30,36)Polycyclic presentation:
2^9|a^2=e,b^2=ef,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72252906, #72256865, #72256876 |
| maximal normal subgroup of | #72252906, #72256865, #72256876 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328681);