Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #495737 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21452 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33541 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1017 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8,12,23)(3,13)(4,16)(5,6)(9,24,26,31)(10,22,11,17)(14,29)(15,20)(18,19)(21,27,32,25);(1,2)(3,9)(4,17)(5,11)(6,10)(7,12)(8,19)(13,25)(14,24)(15,26)(16,23)(18,22)(20,27)(21,30)(28,32)(29,31);(1,3)(2,9)(4,14)(5,20)(6,13)(7,15)(8,21)(10,25)(11,27)(12,26)(16,28)(17,24)(18,29)(19,30)(22,31)(23,32);(2,10,12,11)(4,18)(8,22,23,17)(9,25,26,27)(14,29)(16,19)(21,31,32,24)(28,30)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=dh,e^b=ehi,g^b=gi,h^b=hi,f^c=fi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72214039, #72237280, C2×#495737, … |
| maximal normal subgroup of | #72214039, #72237280, C2×#495737, … |
| maximal quotient of | #72216443, #72225271, #72225273, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402933);