Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388123 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14516 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,6,13,7,11)(3,14,17,25,18,9,16,8)(4,19,21,28,22,27,23,10)(15,24,31,26,30,32,20,29);(1,3,5,17,6,18,7,16)(2,8,12,14,13,25,11,9)(4,20,23,30,22,31,21,15)(10,26,27,24,28,29,19,32);(1,3,6,18)(2,9)(4,15,22,30)(5,16,7,17)(8,11)(10,26)(12,25)(13,14)(19,24)(20,23,31,21)(27,32)(28,29);(1,4,6,22)(2,10)(3,15,18,30)(5,23,7,21)(8,24)(9,26)(11,19)(12,27)(13,28)(14,29)(16,31,17,20)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=efi,c^2=efg,d^2=efg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237047, #72237058, #72237063, … |
| maximal normal subgroup of | #72237047, #72237058, #72237063, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295322);