Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420064 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2495 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
696 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)(5,6,12,15,11,16,10,28)(7,9,18,13,19,27,29,26)(8,20,23,32,25,14,24,33)(17,22,21,36,35,31,34,30);(5,7,10,29,11,19,12,18)(6,13,28,9,16,26,15,27)(8,21,23,35,25,34,24,17)(14,31,33,30,20,22,32,36);(5,8,10,23,11,25,12,24)(6,14,28,33,16,20,15,32)(7,17,29,21,19,35,18,34)(9,22,26,36,27,31,13,30);(2,4)(5,9,10,26,11,27,12,13)(6,7,15,18,16,19,28,29)(8,22,23,36,25,31,24,30)(14,17,32,34,20,35,33,21)Polycyclic presentation:
2^9|a^2=efgh,b^2=ef,c^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358929);