Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499514 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23708 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13323 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,11,21,9,29,28,6,10,20,27,5,12,19,8)(3,13,15,22,16,25,31,32,17,7,18,24,14,23,30,26);(1,3)(2,7)(4,18)(5,14)(6,17)(8,26)(9,23)(10,13)(11,22)(12,25)(15,20)(16,21)(19,31)(24,27)(28,32)(29,30);(2,8)(3,14)(4,19)(5,21)(7,22)(9,27)(10,28)(11,12)(13,24)(15,18)(16,17)(20,29)(23,26)(25,32);(2,9,10,12)(4,20)(7,23,13,25)(8,27,28,11)(15,18)(19,29)(22,26,24,32)(30,31)Polycyclic presentation:
2^9|a^2=f,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fh,g^c=gi,h^c=hi,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237280, C2×#499514, #72320520, … |
| maximal normal subgroup of | #72237280, C2×#499514, #72320520, … |
| maximal quotient of | C2×#499514, #72250226, #72250230, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406710);