Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#379 |
| prime radical | #62748 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2274 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4,10,12)(5,7)(6,8)(9,11,15,16)(13,14)(17,18,21,25,22,26,20,30)(19,27,28,23,29,24,31,32);(1,3)(2,4)(5,9)(6,10)(7,11)(8,12)(13,15)(14,16)(17,19)(18,23)(20,31)(21,28)(22,29)(24,25)(26,32)(27,30);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,16)(14,15)(17,19)(18,24)(20,28)(21,31)(22,29)(23,25)(26,27)(30,32);(1,5,6,13)(2,7,8,14)(3,9,10,15)(4,11,12,16)Polycyclic presentation:
2^9|a^2=g,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94494);