Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493492 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20585 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62836 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #3498 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2237 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)(3,12)(4,16,18,9)(5,10)(6,11)(7,13,23,28)(8,17,24,19)(14,21)(15,26)(20,25)(22,27,32,29)(30,31);(1,3,6,15)(2,7,11,23)(4,13,18,28)(5,14,20,30)(8,21,24,31)(9,12,16,26)(10,22,25,32)(17,27,19,29);(1,4)(2,8)(3,13)(5,19)(6,18)(7,21)(9,10)(11,24)(12,22)(14,29)(15,28)(16,25)(17,20)(23,31)(26,32)(27,30);(5,20)(8,24)(10,25)(14,30)(17,19)(21,31)(22,32)(27,29)Polycyclic presentation:
2^9|a^2=g,b^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,g^b=gh,d^c=dh,f^c=fi,g^c=gi,f^d=fi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72234103, #72260467, #72260564 |
| maximal normal subgroup of | #72234103, #72260467, #72260564 |
| maximal quotient of | #72234078, #72260917, #72261611, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400692);