Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#381 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4126 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)(7,15)(8,19,21,12)(9,22)(10,14)(11,18)(13,23)(16,29,30,27)(17,28)(20,26,35,34)(24,31,33,36)(25,32);(5,7)(6,11,14,25)(8,16)(9,18)(10,17)(12,24,19,33)(13,15,22,28)(20,31)(21,30)(23,32)(26,27,34,29)(35,36);(2,4)(5,8)(6,12)(7,16)(9,20)(10,21)(11,24)(13,26)(14,19)(15,27)(17,30)(18,31)(22,34)(23,35)(25,33)(28,29)(32,36);(7,17)(11,25)(15,28)(16,30)(18,32)(24,33)(27,29)(31,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=di,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72237175, #72237176, #72237183, … |
| maximal normal subgroup of | #72237175, #72237176, #72237183, … |
| maximal quotient of | #72239042, #72239050, #72280477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260520);