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 |
| prime radical | C2×#3584 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3109 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 3109 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,12)(7,15,17,33)(8,19,23,35)(10,25,22,14)(11,27,28,16)(13,30,21,24)(18,34,26,31)(20,32,36,29);(5,7)(6,11)(8,20)(9,17)(10,26)(12,28)(13,31)(14,32)(15,19)(16,24)(18,22)(21,34)(23,36)(25,29)(27,30)(33,35);(5,6)(7,11)(8,21)(9,12)(10,14)(13,23)(15,27)(16,33)(17,28)(18,29)(19,30)(20,34)(22,25)(24,35)(26,32)(31,36);(2,4)(6,12)(11,28)(13,21)(14,25)(15,33)(19,35)(29,32)(31,34)Polycyclic presentation:
2^9|a^2=i,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 | #72250086, #72250096, #72250120 |
| maximal normal subgroup of | #72250086, #72250096, #72250120 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418950);