Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476293 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1263 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,10,11,24,15)(7,16,17,26,20,29,19,12)(8,21,22,27,23,31,25,13)(18,30,34,32,35,28,33,36);(1,2)(3,4)(6,11)(7,17)(8,22)(9,24)(12,16)(13,21)(19,20)(23,25)(26,29)(27,31)(28,30)(33,34);(2,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,26)(13,30)(14,29)(15,16)(17,24)(21,36)(22,33)(23,35)(25,34)(27,28)(31,32);(5,8,10,23)(6,13,11,27)(7,18,20,35)(9,25,24,22)(12,28,26,30)(14,31,15,21)(16,32,29,36)(17,33,19,34)Polycyclic presentation:
2^9|a^2=f,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,f^d=fi,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, 383490);