Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11769 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1505 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C2×#469 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
474 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 474 conjugacy classes of subgroups — too many to draw. Only its 104 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 — 3 generators on 36 points:
(1,2,4,3)(6,12,14,28)(8,21)(9,10)(13,27,30,29)(15,31,26,22)(16,32,33,36)(17,34)(19,20)(23,35)(24,25);(5,6)(7,16,18,33)(8,22,24,31)(9,26)(10,15)(11,14)(12,25,28,21)(13,20,30,19)(17,27)(23,29)(32,34)(35,36);(1,3,4,2)(5,7)(6,13,14,30)(8,23)(9,19)(10,20)(11,18)(12,27,28,29)(15,16,26,33)(17,24)(21,35)(22,36,31,32)(25,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,g^b=gi,d^c=di,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281512, #72307234, #72307323, … |
| maximal normal subgroup of | #72281512, #72307234, #72307323, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36028);