Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2404 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1051 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1051 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,18)(14,19)(17,21)(20,31)(22,29)(23,40)(24,32)(25,33)(26,35)(27,36)(28,37)(30,39)(34,38);(9,11)(10,15)(12,23,24,39)(13,22)(14,21)(16,30,32,40)(17,19)(18,29)(20,25,37,26)(27,38)(28,33,31,35)(34,36);(2,5)(3,8)(6,7)(9,12)(10,16)(11,20)(13,26)(14,24)(15,28)(17,31)(18,35)(19,32)(21,37)(22,39)(23,38)(25,27)(29,40)(30,34)(33,36);(9,11)(10,17)(12,20)(13,22)(14,21)(15,19)(16,31)(18,34)(23,25)(24,37)(26,39)(27,38)(28,32)(29,36)(30,35)(33,40)Polycyclic presentation:
2^9|b^2=e,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,f^b=fh,e^c=eh,f^c=fi,g^c=gh,f^d=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279995, #72280537, #72280571 |
| maximal normal subgroup of | #72279995, #72280537, #72280571 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 158996);