Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393906 |
| 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 | #2598 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,13)(8,22,23,36)(9,15,25,29)(10,16)(11,27)(12,17)(14,24,30,19)(18,35,26,34)(20,31)(21,33)(28,32);(5,7,10,20,12,21,11,28)(6,13,16,31,17,33,27,32)(8,9,19,18,23,25,24,26)(14,34,36,15,30,35,22,29);(5,8,10,19,12,23,11,24)(6,14,16,36,17,30,27,22)(7,18,20,25,21,26,28,9)(13,29,31,34,33,15,32,35);(1,2,3,4)(5,9,10,18,12,25,11,26)(6,15,16,35,17,29,27,34)(7,19,20,23,21,24,28,8)(13,30,31,22,33,14,32,36)Polycyclic presentation:
2^9|a^2=e,b^2=ef,c^2=ef,d^2=gh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321724);