Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419839 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
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 | |
1253 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,11,13,10,25)(7,16,17,27,20,31,19,12)(8,14,22,30,24,28,23,32)(18,29,21,36,35,33,34,26);(1,2)(3,4)(5,7)(6,12)(8,21)(9,19)(10,17)(11,20)(13,27)(14,29)(15,31)(16,25)(18,22)(23,35)(24,34)(26,30)(28,33)(32,36);(2,4)(6,13)(7,17)(9,10)(12,16)(14,28)(18,34)(19,20)(21,35)(22,23)(26,33)(27,31)(29,36);(5,8)(6,14,13,28)(7,18)(9,23)(10,22)(11,24)(12,26,27,36)(15,32,25,30)(16,33,31,29)(17,34)(19,21)(20,35)Polycyclic presentation:
2^9|a^2=e,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334638);