Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454982 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23445 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1559 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)(7,11,17,25)(8,12)(9,14)(10,13)(15,21)(16,27,31,23)(18,24,33,26)(19,29)(20,30)(22,28)(32,35,34,36);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,27)(13,24)(14,25)(15,26)(19,31)(20,32)(21,33)(22,34)(23,29)(28,36)(30,35);(2,4)(5,8)(6,12)(7,16)(9,19)(10,22)(11,23)(13,28)(14,29)(15,30)(17,31)(18,34)(20,21)(24,35)(25,27)(26,36)(32,33);(6,13,14,15)(8,19)(11,24,25,26)(12,28,29,30)(16,31)(20,22)(23,35,27,36)(32,34)Polycyclic presentation:
2^9|a^2=f,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=de,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285925, #72286224, #72286267, … |
| maximal normal subgroup of | #72285925, #72286224, #72286267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362179);