Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499603 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23764 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#472 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
444 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)(3,14)(4,10)(5,9)(6,12)(7,13)(8,17)(11,22)(15,31)(16,27)(18,26)(19,25)(20,28)(21,29)(23,32)(24,30);(1,3,5,18,7,17,22,20)(2,8,11,26,13,14,9,28)(4,16,6,23,21,19,24,15)(10,25,30,32,29,27,12,31);(2,9)(3,15)(4,6)(5,22)(8,25)(10,29)(11,13)(14,27)(16,20)(17,23)(18,19)(21,24)(26,31)(28,32);(1,4,5,6,7,21,22,24)(2,10,9,12,13,29,11,30)(3,16,18,23,17,19,20,15)(8,25,28,31,14,27,26,32)Polycyclic presentation:
2^9|b^2=f,d^2=eh,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,g^b=gh,h^b=hi,d^c=dh,f^c=fi,g^c=ghi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237280, C2×#499603, #72320520, … |
| maximal normal subgroup of | #72237280, C2×#499603, #72320520, … |
| maximal quotient of | C2×#499603, #72239848, #72239850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406799);