Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476659 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
672 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)(5,6,9,14,11,16,10,15)(7,12,18,30,20,26,19,27)(8,21,23,28,22,29,24,13)(17,31,32,25,33,35,34,36);(5,7)(6,12)(8,17,22,33)(9,18)(10,19)(11,20)(13,25,28,36)(14,30)(15,27)(16,26)(21,35,29,31)(23,32,24,34);(5,8,9,24,11,22,10,23)(6,13,14,29,16,28,15,21)(7,17,19,32,20,33,18,34)(12,25,27,35,26,36,30,31);(2,4)(6,14)(7,18)(8,22)(9,10)(12,26)(13,21)(15,16)(17,32)(19,20)(28,29)(31,35)(33,34)Polycyclic presentation:
2^9|a^2=efg,b^2=e,c^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,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, 385973);