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 | #33740 | C2 |
| 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 | |
200 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 200 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)(4,22,25,8)(5,27)(6,29)(7,14)(9,19)(10,23,39,48)(11,28)(12,24)(13,30)(16,43)(17,40)(18,49,50,59)(20,53)(21,36)(26,34,58,41)(31,51,55,63)(32,56,33,57)(35,52,61,64)(37,47)(38,46)(42,45,60,62)(44,54);(2,8)(3,16)(4,23)(5,24)(6,30)(9,35)(10,12)(11,39)(13,41)(14,22)(15,42)(17,28)(18,50)(19,20)(21,47)(25,48)(26,57)(27,33)(29,34)(31,53)(32,40)(36,60)(37,49)(38,61)(43,59)(44,55)(45,51)(46,54)(56,58)(62,63);(1,3,17,46,6,20,24,47,7,21,5,19,30,54,28,16)(2,9,12,15,13,37,40,44,14,38,11,36,29,43,27,53)(4,18,48,63,26,52,57,62,25,50,23,51,58,64,56,45)(8,31,33,59,34,60,39,61,22,55,32,49,41,42,10,35)Polycyclic presentation:
2^9|a^2=e,c^2=fi,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, #72321005, … |
| maximal normal subgroup of | #72286130, #72320907, #72321005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57994);