Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198171 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6,10,15)(4,11)(5,8)(7,13)(9,16,12,14)(17,18,22,26,23,19,21,25)(20,28,30,24,31,32,29,27);(1,3,5,10)(2,6)(4,12,13,9)(7,16)(8,15)(11,14)(17,19,23,18)(20,27,31,24)(21,25,22,26)(28,30,32,29);(4,13)(7,11)(9,12)(14,16)(17,20,23,31)(18,24,19,27)(21,30,22,29)(25,32,26,28);(1,4,5,13)(2,7,8,11)(3,9,10,12)(6,14,15,16)(18,19)(24,27)(25,26)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=hi,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,f^c=fhi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72315839, #72315842, #72315907, … |
| maximal normal subgroup of | #72315839, #72315842, #72315907, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105683);