Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2408 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1863 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,13)(8,21)(9,17)(10,15)(11,16)(12,19)(14,25)(18,27)(20,30)(22,29)(23,33)(24,32)(26,28)(31,35)(34,36);(1,3,4,2)(5,7)(6,12)(8,22)(9,26)(10,19)(11,20)(13,17)(14,31)(15,30)(16,28)(18,23)(21,36)(24,35)(25,34)(27,32)(29,33);(2,3)(5,7)(6,13)(8,18)(9,26)(10,19)(11,20)(12,15)(14,29)(16,30)(17,28)(21,27)(22,25)(23,35)(24,34)(31,33)(32,36);(2,3)(5,8)(6,14)(7,18)(9,25)(10,23)(11,24)(12,27)(13,29)(15,21)(16,33)(17,32)(19,35)(20,34)(22,26)(28,36)(30,31)Polycyclic presentation:
2^9|b^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,e^c=ei,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328036);