Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455572 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1188 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)(7,17)(8,22)(9,15)(10,13)(11,16)(12,18)(14,24)(19,27)(20,26)(21,28)(23,30)(25,29)(31,33)(32,36)(34,35);(1,2)(3,4)(5,7,10,20,11,21,9,18)(6,12,15,28,16,26,13,17)(8,19,23,33,25,34,24,32)(14,27,29,36,30,35,22,31);(2,4)(6,13)(7,18)(8,23)(9,10)(12,26)(15,16)(20,21)(22,29)(24,25)(27,31)(32,33)(35,36);(5,8)(6,14)(7,19)(9,24)(10,23)(11,25)(12,27)(13,22)(15,29)(16,30)(17,31)(18,32)(20,33)(21,34)(26,35)(28,36)Polycyclic presentation:
2^9|b^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362768);