Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #497430 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #22436 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #62636 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1765 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
954 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)(3,13,16,29)(4,18)(5,10)(6,12)(7,14,25,31)(8,15,26,17)(9,21)(11,23)(19,30,32,24)(20,22)(27,28);(1,3)(2,7)(4,19)(5,15)(6,16)(8,9)(10,24)(11,30)(12,25)(13,20)(14,27)(17,23)(18,29)(21,31)(22,32)(26,28);(1,3)(2,8)(4,14)(5,17)(6,16)(7,9)(10,29)(11,13)(12,26)(15,23)(18,24)(19,27)(20,30)(21,32)(22,31)(25,28);(1,2)(3,8)(4,20)(5,10)(6,12)(7,19)(9,27)(11,23)(13,15)(14,30)(16,26)(17,29)(18,22)(21,28)(24,31)(25,32)Polycyclic presentation:
2^9|a^2=g,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=de,e^b=ehi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=ehi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72214039, #72285481, #72285521, … |
| maximal normal subgroup of | #72214039, #72285481, #72285521, … |
| maximal quotient of | #72216444, #72227130, #72227132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404626);