Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | C2×D4 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×C22⋊C16 |
| prime radical | #532 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
243 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,17,13,18,25,35,14,19,26,36,28,38,47,57,15,20,27,37,29,39,48,58,30,40,49,59,50,60,65,70)(11,16,22,32,23,33,44,54,24,34,45,55,46,56,64,69,21,31,41,51,42,52,61,66,43,53,62,67,63,68,71,72);(2,6)(3,4)(7,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,21)(17,32)(18,33)(19,34)(20,31)(25,44)(26,45)(27,41)(28,46)(29,42)(30,43)(35,54)(36,55)(37,51)(38,56)(39,52)(40,53)(47,64)(48,61)(49,62)(50,63)(57,69)(58,66)(59,67)(60,68)(65,71)(70,72);(1,3,5,4)(2,7,8,6)(11,21)(16,31)(22,41)(23,42)(24,43)(32,51)(33,52)(34,53)(44,61)(45,62)(46,63)(54,66)(55,67)(56,68)(64,71)(69,72)Polycyclic presentation:
2^9|a^2=e,c^2=f,d^2=f,e^2=g,g^2=h,h^2=i,b^a=bd,d^a=df,c^b=cfi,d^b=df152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57296);