Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359665 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13867 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3885 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,12,5,11)(3,13,15,8,16,24,21,25)(4,17,18,9,20,27,19,26)(14,23,30,32,31,22,29,28);(1,3)(2,8)(4,14)(5,21)(6,15)(7,16)(9,23)(10,24)(11,13)(12,25)(17,28)(18,30)(19,29)(20,31)(22,26)(27,32);(1,4)(2,9)(3,14)(5,18)(6,19)(7,20)(8,22)(10,17)(11,27)(12,26)(13,28)(15,29)(16,31)(21,30)(23,25)(24,32);(1,4)(2,9)(3,14)(5,19)(6,18)(7,20)(8,23)(10,27)(11,17)(12,26)(13,28)(15,30)(16,31)(21,29)(22,25)(24,32)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72235139, #72251422, #72251427, … |
| maximal normal subgroup of | #72235139, #72251422, #72251427, … |
| maximal quotient of | #72235212, #72286774, #72287117 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266863);