Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480314 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1782 |
| prime radical | #62686 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2221 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)(7,13,21,26)(8,17,23,29)(14,25)(15,20)(19,24)(22,27,31,32)(28,30);(1,3)(2,7)(4,13)(5,15)(6,14)(8,20)(9,12)(10,22)(11,21)(16,25)(17,27)(18,26)(19,28)(23,30)(24,31)(29,32);(1,4)(2,8,11,23)(3,13)(5,17)(6,18)(7,20,21,30)(9,10,16,24)(12,22,25,31)(14,26)(15,27)(19,29)(28,32);(3,14)(7,21)(12,25)(13,26)(15,28)(20,30)(22,31)(27,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,g^b=gh,g^c=gi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306736 |
| maximal normal subgroup of | #72306736 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400640);