Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150243 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C23 | C2×SD32 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
215 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 215 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(2,6,7,8)(9,10,13,18,15,21,34,20)(11,22,25,53,23,52,27,61)(12,28,31,62,33,51,67,63)(14,35,30,19,37,47,60,49)(16,38,40,54,39,57,50,24)(17,42,44,56,46,68,45,32)(26,59,58,70,55,71,36,43)(29,48,65,69,64,72,66,41);(1,2)(3,6)(4,8)(5,7)(9,11,15,23)(10,16,21,39)(12,29,33,64)(13,25,34,27)(14,36,37,58)(17,43,46,70)(18,40,20,50)(19,48,49,72)(22,51,52,28)(24,56,54,32)(26,60,55,30)(31,65,67,66)(35,41,47,69)(38,68,57,42)(44,59,45,71)(53,63,61,62);(1,3,5,4)(2,6,7,8)(11,23)(16,39)(22,52)(24,54)(25,27)(26,55)(29,64)(36,58)(38,57)(40,50)(41,69)(43,70)(48,72)(53,61)(59,71)(65,66)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=fgh,d^2=gh,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gh,h^a=hi,c^b=ci,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57584);