Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | #1999 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(10,13)(11,15)(14,16)(17,18,21,25,22,27,28,26)(19,23,20,31,30,24,29,32);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19,28,29,22,30,21,20)(18,23,26,32,27,24,25,31);(3,8)(5,7)(9,11,13,15,12,16,10,14)(17,20,21,30,22,29,28,19)(18,24,26,31,27,23,25,32);(2,6)(5,7)(9,10,12,13)(11,14,16,15)(17,21,22,28)(18,25,27,26)(19,29,30,20)(23,32,24,31)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=f,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=di,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72320417, C2×#218804, #72343117, … |
| maximal normal subgroup of | #72320417, C2×#218804, #72343117, … |
| maximal quotient of | #72320588, C2×#218804, #72343422, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 126000);