Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151002 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11975 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | C23 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
172 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,5,13,7,34,4,24,6,33,25,38,26,47,27,12,9,16,32,10,36,15,29,46,8,14,31,45,35,42,28,43)(3,17,19,44,21,41,56,60,20,40,57,61,53,62,55,39,23,11,30,37,59,50,18,51,22,49,52,63,58,64,54,48);(2,10,16,13)(4,25)(5,27)(6,8)(7,35)(11,37,17,44)(12,15,43,34)(14,45,33,38)(18,52)(19,55)(20,22)(21,58)(24,42,46,47)(26,36)(28,32)(29,31)(30,54)(39,41,48,50)(40,63,49,61)(51,64,60,62)(53,59)(56,57);(1,3)(2,11)(4,18)(5,30)(6,20)(7,21)(8,22)(9,23)(10,37)(12,39)(13,44)(14,40)(15,41)(16,17)(19,32)(24,60)(25,52)(26,53)(27,54)(28,55)(29,56)(31,57)(33,49)(34,50)(35,58)(36,59)(38,61)(42,62)(43,48)(45,63)(46,51)(47,64)Polycyclic presentation:
2^9|a^2=e,b^2=fh,d^2=ghi,e^2=fgh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efg,f^b=fi,g^b=gh,h^b=hi,e^c=ei44 conjugacy classes — view the full character table.
| maximal subgroup of | #72286285, #72343284, #72343286, … |
| maximal normal subgroup of | #72286285, #72343284, #72343286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58198);