Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125508 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#102 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
326 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 326 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 56 points:
(1,2,4,7,5,6,3,8)(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23)(25,26)(27,35)(28,39)(29,44)(30,32)(31,36)(33,42)(34,41)(37,55)(38,52)(40,50)(43,51)(45,49)(46,54)(47,56)(48,53);(2,6)(3,4)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,30,38)(26,31,32,46)(28,40,43,56)(29,45,41,37)(33,49,48,55)(34,50,44,47)(35,51,52,39)(36,53,54,42);(1,3,5,4)(2,7,6,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,32)(31,46)(33,48)(34,44)(35,52)(39,51)(47,50)(49,55)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=fgi,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=cfg,d^a=df,f^a=fi,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,g^b=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32738);