Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C22 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×C4×C22⋊C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2398 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)(7,15,17,11)(8,12)(9,14)(10,13)(16,27,29,22)(18,28,31,23)(19,24)(20,25)(21,26)(30,35,36,33)(32,34);(1,2,3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,22)(13,15)(14,23)(19,29)(20,30)(21,31)(24,27)(25,33)(26,28)(32,36)(34,35);(5,8,9,20,10,19,21,32)(6,12,14,25,13,24,26,34)(7,16,18,30,17,29,31,36)(11,22,23,33,15,27,28,35);(8,19)(12,24)(16,29)(20,32)(22,27)(25,34)(30,36)(33,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,h^2=i,b^a=be,e^a=ei,f^b=fi,d^c=di164 conjugacy classes — view the full character table.
| maximal subgroup of | #72251065, #72251086, #72251096, … |
| maximal normal subgroup of | #72251065, #72251086, #72251096, … |
| maximal quotient of | #72314706 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260353);