Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60470 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#121 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×C4×#121 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2599 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,12,8,13,6,23)(3,14,5,11,16,28,17,9)(4,18,21,10,22,26,20,27)(15,24,19,29,30,25,31,32);(1,3,6,17,8,16,7,5)(2,9,12,14,13,11,23,28)(4,15,20,31,22,30,21,19)(10,24,26,29,27,25,18,32);(1,4)(2,10)(3,15)(5,19)(6,20)(7,21)(8,22)(9,24)(11,25)(12,26)(13,27)(14,29)(16,30)(17,31)(18,23)(28,32);(1,5,7,16,8,17,6,3)(2,11,12,28,13,9,23,14)(4,19,21,30,22,31,20,15)(10,25,26,32,27,24,18,29);(3,16)(5,17)(9,11)(14,28)(15,30)(19,31)(24,25)(29,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,e^b=ei,g^c=gi,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72213889, #72214259, #72214632, … |
| maximal normal subgroup of | #72213889, #72214259, #72214632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280117);