Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #374124 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14134 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2504 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)(3,12)(4,8,18,25)(5,9)(6,11)(7,14)(10,20)(13,28,16,27)(15,26)(17,23,31,24)(19,22)(21,29,32,30);(1,3)(2,7)(4,16)(5,19)(6,15)(8,21)(9,12)(10,26)(11,22)(13,18)(14,20)(17,29)(23,27)(24,28)(25,32)(30,31);(1,4)(2,8,11,25)(3,13,15,16)(5,17)(6,18)(7,21)(9,23,10,24)(12,27)(14,29,19,30)(20,31)(22,32)(26,28);(2,9)(5,20)(7,12)(8,23)(10,11)(14,19)(17,31)(21,27)(22,26)(24,25)(28,32)(29,30)Polycyclic presentation:
2^9|a^2=h,c^2=fh,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72240432, #72240443, #72240447, … |
| maximal normal subgroup of | #72240432, #72240443, #72240447, … |
| maximal quotient of | C2×#374124, #72255850, #72256108 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281320);