Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493474 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20585 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #62820 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #3485 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2272 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)(3,8)(4,9)(5,16,17,7)(6,10)(11,20)(12,26,27,18)(13,21)(14,28,29,19)(15,23)(22,25)(24,31,32,30);(2,7)(8,18)(9,19)(10,16)(20,30)(21,26)(22,31)(23,28);(1,3,6,13)(2,8,10,21)(4,11,15,25)(5,12,17,27)(7,18,16,26)(9,22,23,20)(14,24,29,32)(19,31,28,30);(1,4)(2,9)(3,11)(5,14)(6,15)(7,19)(8,20)(10,23)(12,24)(13,25)(16,28)(17,29)(18,30)(21,22)(26,31)(27,32)Polycyclic presentation:
2^9|a^2=g,c^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,g^b=gh,d^c=dh,f^d=fi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72233665, #72240644, #72240648, … |
| maximal normal subgroup of | #72233665, #72240644, #72240648, … |
| maximal quotient of | #72233603, #72233651, #72255659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400672);