Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246927 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1416 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 1416 conjugacy classes of subgroups — too many to draw. Only its 150 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:
(2,5)(3,7)(6,8)(9,10)(11,20)(12,17)(13,25)(14,27)(15,19)(16,21)(18,24)(22,30)(23,32)(26,33)(28,31)(29,34)(35,37)(36,39)(38,40);(3,7)(5,8)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,32)(18,30)(19,31)(20,27)(24,36)(25,35)(26,38)(29,37)(33,39)(34,40);(1,2,4,6)(3,8,7,5)(9,12,15,14)(10,17,19,27)(11,21,23,28)(13,24,26,29)(16,20,31,32)(18,33,34,25)(22,36,38,37)(30,35,40,39);(3,7)(5,8)(9,13,15,26)(10,18,19,34)(11,22,23,38)(12,24,14,29)(16,30,31,40)(17,33,27,25)(20,35,32,39)(21,36,28,37)Polycyclic presentation:
2^9|c^2=hi,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,f^c=fi,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72283955, #72317573, #72317975, … |
| maximal normal subgroup of | #72283955, #72317573, #72317975, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 154134);