Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150977 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11952 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | #2218 | C4 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
191 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 191 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:
(1,2,5,13,33,15,4,24,6,11,34,44,27,45,28,12,9,16,32,47,7,36,26,43,8,14,29,50,35,46,30,48)(3,17,19,10,38,40,56,41,20,53,25,51,37,62,31,63,23,49,58,42,21,61,18,55,22,54,59,39,60,64,57,52);(1,3)(2,10)(4,25)(5,31)(6,22)(7,37)(8,20)(9,23)(11,39)(12,40)(13,49)(14,51)(15,52)(16,42)(17,47)(18,29)(19,28)(21,27)(24,62)(26,59)(30,58)(32,57)(33,60)(34,56)(35,38)(36,63)(41,46)(43,64)(44,53)(45,55)(48,61)(50,54);(2,11,16,14)(4,26)(5,32)(10,39,42,51)(12,24,48,43)(13,50,47,44)(15,45,36,46)(17,53,49,54)(18,56)(19,58)(25,59)(28,30)(29,34)(31,57)(40,62,61,64)(41,52,55,63)Polycyclic presentation:
2^9|a^2=e,c^2=fh,d^2=gh,e^2=fghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gh,h^b=hi,d^c=di,e^c=ei50 conjugacy classes — view the full character table.
| maximal subgroup of | #72286285, #72287335, #72287365, … |
| maximal normal subgroup of | #72286285, #72287335, #72287365, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58175);