Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476654 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| 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 | |
1008 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)(7,18)(8,22)(9,14,26,33)(10,16)(11,15)(12,17)(13,20)(19,31,23,36)(21,29)(24,30,35,28)(25,34,27,32);(1,2)(3,4)(5,7,12,8)(6,13,17,29)(9,23,26,19)(10,21,11,20)(14,30,33,28)(15,22,16,18)(24,25,35,27)(31,34,36,32);(2,4)(5,8,12,7)(6,13,17,29)(9,24,26,35)(10,20,11,21)(14,31,33,36)(15,22,16,18)(19,25,23,27)(28,34,30,32);(5,9,10,25,12,26,11,27)(6,14,15,32,17,33,16,34)(7,19,20,35,8,23,21,24)(13,28,18,31,29,30,22,36)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=i,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,e^c=ei,f^c=fi,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, 383853);