Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6371542 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#386 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6255 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,11,7,9,5,20)(3,12,16,21,15,26,14,8)(4,10,19,24,17,23,18,30)(13,25,29,31,27,32,28,22);(1,3)(2,8,9,21)(4,13)(5,14)(6,16)(7,15)(10,22,23,31)(11,12,20,26)(17,27)(18,28)(19,29)(24,25,30,32);(2,9)(5,6)(8,21)(10,23)(14,16)(18,19)(22,31)(28,29);(1,4)(2,10)(3,13)(5,18)(6,19)(7,17)(8,22)(9,23)(11,24)(12,25)(14,28)(15,27)(16,29)(20,30)(21,31)(26,32);(4,17)(10,23)(13,27)(18,19)(22,31)(24,30)(25,32)(28,29)Polycyclic presentation:
2^9|a^2=h,b^2=fh,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,g^b=gi,f^c=fi,h^c=hi,e^d=ei113 conjugacy classes — view the full character table.
| maximal subgroup of | #72251104, #72251110, #72251414, … |
| maximal normal subgroup of | #72251104, #72251110, #72251414, … |
| maximal quotient of | #72255843, #72256023, C2×#6371547, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278743);