Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C22×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
17262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,18,11)(8,12)(9,13)(10,14)(16,27,29,22)(17,28,31,23)(19,24)(20,25)(21,26)(30,35,36,33)(32,34);(5,7)(6,11)(8,16)(9,17)(10,18)(12,22)(13,23)(14,15)(19,29)(20,30)(21,31)(24,27)(25,33)(26,28)(32,36)(34,35);(2,4)(5,8)(6,12)(7,16)(9,20)(10,19)(11,22)(13,25)(14,24)(15,27)(17,30)(18,29)(21,32)(23,33)(26,34)(28,35)(31,36);(8,19)(12,24)(16,29)(20,32)(22,27)(25,34)(30,36)(33,35);(5,9,10,21)(6,13,14,26)(7,17,18,31)(8,20,19,32)(11,23,15,28)(12,25,24,34)(16,30,29,36)(22,33,27,35)Polycyclic presentation:
2^9|a^2=h,e^2=i,b^a=bf,c^a=cg,f^a=fi,h^b=hi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239305, #72255650, #72255660, … |
| maximal normal subgroup of | #72239305, #72255650, #72255660, … |
| maximal quotient of | #72239232, #72239238, #72303967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277888);