Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476297 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1009 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)(5,6,11,17,12,18,10,16)(7,19,22,13,9,29,21,15)(8,23,26,33,27,32,28,14)(20,34,25,36,24,30,35,31);(5,7,12,9)(6,13,18,15)(8,24,27,20)(10,22,11,21)(14,30,33,34)(16,29,17,19)(23,36,32,31)(25,28,35,26);(5,8,10,26,12,27,11,28)(6,14,16,23,18,33,17,32)(7,20,21,25,9,24,22,35)(13,30,19,31,15,34,29,36);(2,4)(5,9,12,7)(6,15,18,13)(8,25,27,35)(10,21,11,22)(14,31,33,36)(16,19,17,29)(20,28,24,26)(23,30,32,34)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=e,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383679);