Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384436 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
960 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,24)(3,14,16,9,17,11,5,23)(4,18,21,29,22,10,25,30)(15,28,20,26,19,31,32,27);(1,3)(2,9)(4,19)(5,6)(7,16)(8,17)(10,27)(11,12)(13,23)(14,24)(15,22)(18,26)(20,25)(21,32)(28,30)(29,31);(1,4,6,21,8,22,7,25)(2,10,24,30,13,18,12,29)(3,15,16,32,17,19,5,20)(9,26,11,28,23,27,14,31);(1,5,7,17,8,16,6,3)(2,11,24,23,13,14,12,9)(4,20,25,19,22,32,21,15)(10,28,30,27,18,31,29,26)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^c=gi,h^c=hi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72339024, #72339027, #72339034 |
| maximal normal subgroup of | #72339024, #72339027, #72339034 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291635);