Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150390 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11744 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C2×M32 | C23 |
| prime radical | C23 | #509 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
210 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,15)(4,10)(5,24,27,39)(6,30)(7,32)(8,14)(9,17,35,53)(11,28,42,51)(12,23)(13,21)(16,41)(18,55)(19,57)(20,48)(22,59,26,43)(25,44,61,38)(29,63,58,60)(31,40)(33,47)(34,52,36,49)(37,62,45,54)(46,50)(56,64);(1,3,8,20)(2,9,14,35)(4,18,23,56)(5,25,27,61)(6,16,31,50)(7,33,21,19)(10,38,12,44)(11,41,42,46)(13,45,32,37)(15,24,48,39)(17,51,53,28)(22,54,26,62)(29,49,58,52)(30,36,40,34)(43,55,59,64)(47,63,57,60);(1,4,21,31,8,23,7,6)(2,10,13,40,14,12,32,30)(3,16,33,56,20,50,19,18)(5,26,58,28,27,22,29,51)(9,34,37,38,35,36,45,44)(11,24,43,60,42,39,59,63)(15,41,47,64,48,46,57,55)(17,52,62,25,53,49,54,61)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=fgi,d^2=ghi,f^2=h,g^2=h,h^2=i,b^a=bd,d^a=df,f^a=fh,g^a=ghi,h^a=hi,c^b=ch,d^b=dg,e^b=efghi,f^b=fhi,g^b=ghi,h^b=hi,d^c=di,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57588);