Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343418 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62773 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,8,10,23)(4,18)(5,19)(9,22,25,31)(11,24,12,20)(14,29)(15,28)(26,32,27,30);(1,2)(3,9)(4,8)(5,20)(6,11)(7,12)(10,21)(13,25)(14,22)(15,30)(16,26)(17,27)(18,24)(19,23)(28,31)(29,32);(1,3,6,16)(2,9,11,26)(4,14,18,29)(5,15,19,28)(7,17,21,13)(8,22,24,32)(10,25,12,27)(20,30,23,31);(3,13)(9,25)(14,28)(15,29)(16,17)(22,31)(26,27)(30,32);(1,4,7,5)(2,8,12,20)(3,14,17,15)(6,18,21,19)(9,22,27,30)(10,23,11,24)(13,28,16,29)(25,31,26,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^2=i,f^2=i,b^a=bf,e^a=eh,f^a=fh,f^b=fi,g^b=ghi,d^c=dhi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72232826, #72233035, #72233129, … |
| maximal normal subgroup of | #72232826, #72233035, #72233129, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250709);