Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511753 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3138 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 3138 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,11)(8,18)(9,21)(12,26)(13,25)(14,29)(16,31)(17,30)(19,23)(20,34)(22,35)(24,27)(28,36)(32,33);(5,6)(7,12)(8,19)(9,23)(10,13)(11,18)(14,21)(15,27)(16,32)(17,33)(20,36)(22,29)(24,34)(25,35)(26,31)(28,30);(5,7)(6,12)(8,20)(9,17)(10,15)(11,24)(13,27)(14,28)(16,22)(18,34)(19,36)(21,30)(23,33)(25,26)(29,32)(31,35);(2,4)(7,15)(12,27)(16,20)(17,30)(24,26)(28,33)(31,34)(32,36)Polycyclic presentation:
2^9|e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,h^a=hi,e^b=eh,f^b=fi,h^b=hi,d^c=di,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72249968, #72250086, #72250090, … |
| maximal normal subgroup of | #72249968, #72250086, #72250090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418949);