Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460861 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| 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 | |
820 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)(5,6,11,16)(7,17)(8,13)(9,15,10,14)(12,20)(18,33,35,29)(19,28)(21,27)(22,36,34,26)(23,31)(24,30)(25,32);(5,7,9,19,11,21,10,20)(6,12,14,27,16,28,15,17)(8,22,23,35,25,34,24,18)(13,29,30,36,32,33,31,26);(5,8,9,23,11,25,10,24)(6,13,14,30,16,32,15,31)(7,18,19,22,21,35,20,34)(12,26,27,29,28,36,17,33);(2,4)(6,14,16,15)(8,23,25,24)(12,27,28,17)(18,22,35,34)Polycyclic presentation:
2^9|a^2=efg,b^2=ef,c^2=ef,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371089);