Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493562 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1790 |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #62862 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2513 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)(3,11)(4,8)(5,17,18,9)(6,10)(7,13,20,29)(12,26)(14,25)(15,21)(16,23,30,22)(19,28,31,32)(24,27);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,11)(10,20)(15,27)(16,28)(17,25)(18,29)(21,31)(22,26)(23,24)(30,32);(1,4)(2,8)(3,12)(5,16)(6,15)(7,19)(9,23)(10,21)(11,24)(13,28)(14,27)(17,22)(18,30)(20,31)(25,26)(29,32);(4,15)(8,21)(12,27)(16,30)(19,31)(22,23)(24,26)(28,32)Polycyclic presentation:
2^9|a^2=g,b^a=be,c^a=cf,e^a=eh,h^a=hi,f^b=fhi,g^b=gh,d^c=di,e^c=eh,f^e=fi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72232886, #72233205, #72235130, … |
| maximal normal subgroup of | #72232886, #72233205, #72235130, … |
| maximal quotient of | #72232779, #72233196, #72233200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400758);