Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493411 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1781 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
3407 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)(3,14,17,29)(4,10)(5,11,21,28)(6,12)(7,13)(8,16)(9,15,26,18)(19,22,20,27)(23,30)(24,31)(25,32);(1,3)(2,8)(4,18)(5,16)(6,15)(7,17)(9,22)(10,25)(11,14)(12,23)(13,24)(19,32)(20,30)(21,31)(26,27)(28,29);(1,3)(2,9)(4,15)(5,16)(6,18)(7,17)(8,22)(10,14)(11,25)(12,29)(13,26)(19,30)(20,32)(21,31)(23,28)(24,27);(1,4,7,6)(2,10,13,12)(3,15,17,18)(5,19,21,20)(8,23,24,25)(9,14,26,29)(11,22,28,27)(16,30,31,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,g^b=gh,f^c=fi,g^c=gi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72261398, #72261432, #72261527, … |
| maximal normal subgroup of | #72261398, #72261432, #72261527, … |
| maximal quotient of | #72287199, #72318049, #72318640, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400609);