Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,8,11,16,10,15)(7,17,22,18,21,29,20,13)(9,23,27,14,26,31,25,32)(19,28,24,30,36,34,35,33);(1,2)(3,4)(5,7,10,20,11,21,12,22)(6,13,8,17,16,18,15,29)(9,24,25,19,26,35,27,36)(14,30,31,34,32,33,23,28);(2,4)(5,8,10,16,11,15,12,6)(7,18,20,29,21,13,22,17)(9,14,25,31,26,32,27,23)(19,33,35,28,36,30,24,34);(5,9,11,26)(6,14,16,32)(7,19,21,36)(8,23,15,31)(10,27,12,25)(13,28,18,34)(17,33,29,30)(20,24,22,35)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=e,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383806);