Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2253 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)(5,6,9,12,11,16,10,15)(7,17,20,13,21,27,19,28)(8,14,23,25,24,29,22,30)(18,31,32,26,34,36,33,35);(6,12,16,15)(8,22,24,23)(13,17,28,27)(14,29)(18,32,34,33)(25,30)(26,35)(31,36);(2,4)(5,7)(6,13)(8,18)(9,19)(10,20)(11,21)(12,17)(14,26)(15,27)(16,28)(22,32)(23,33)(24,34)(25,31)(29,35)(30,36);(5,8)(6,14)(7,18)(9,23)(10,22)(11,24)(12,25)(13,26)(15,30)(16,29)(17,31)(19,33)(20,32)(21,34)(27,36)(28,35)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=de,f^c=fi,g^c=gi,e^d=ei,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72224693, #72312219, #72312221, … |
| maximal normal subgroup of | #72224693, #72312219, #72312221, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263305);