Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493594 |
| 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 | |
2501 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)(4,15,17,8)(5,18,19,9)(6,10)(7,14,21,28)(12,24,26,31)(13,25)(16,22)(20,27)(23,30)(29,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,20)(9,11)(10,21)(15,29)(16,27)(17,26)(18,25)(19,28)(22,31)(23,24)(30,32);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,23)(10,15)(11,24)(13,26)(14,27)(18,22)(19,30)(21,29)(25,31)(28,32);(3,13)(7,21)(11,25)(12,26)(14,28)(20,29)(24,31)(27,32)Polycyclic presentation:
2^9|a^2=g,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=di,f^b=fhi,g^b=gh,e^c=eh,g^c=gi,f^e=fi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72232886, #72237472, #72238342, … |
| maximal normal subgroup of | #72232886, #72237472, #72238342, … |
| maximal quotient of | #72232750, #72233297, #72233333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400790);