Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150797 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11858 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C2×M32 | C23 |
| prime radical | #2218 | C22 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
194 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 58 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)(4,21,24,8)(5,26)(6,29)(7,14)(9,18)(10,22,39,55)(11,27)(12,23)(13,30)(16,43)(17,48,49,59)(19,52)(20,36)(25,34,58,41)(28,40)(31,50,54,63)(32,56,33,57)(35,51,61,64)(37,47)(38,46)(42,45,60,62)(44,53);(2,8)(3,16)(4,22)(5,23)(6,30)(9,35)(10,12)(11,39)(13,41)(14,21)(15,42)(17,49)(18,19)(20,47)(24,55)(25,57)(26,33)(27,28)(29,34)(31,52)(32,40)(36,60)(37,48)(38,61)(43,59)(44,54)(45,50)(46,53)(56,58)(62,63);(1,3,5,18,6,19,27,16,7,20,28,46,30,53,23,47)(2,9,11,36,13,37,26,52,14,38,12,15,29,43,40,44)(4,17,22,50,25,51,56,45,24,49,55,63,58,64,57,62)(8,31,32,48,34,60,10,35,21,54,33,59,41,42,39,61)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=ghi,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=ghi,h^a=hi,c^b=cfh,d^b=dg,e^b=efghi,f^b=fhi,g^b=ghi,h^b=hi,e^d=ei44 conjugacy classes — view the full character table.
| maximal subgroup of | #72286130, #72320907, #72321032, … |
| maximal normal subgroup of | #72286130, #72320907, #72321032, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57993);