Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13691 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2206 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C2×#2246 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
368 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 94 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:
(2,4)(5,6,9,14,8,20,22,34,12,15,27,30,23,21,33,35,13,16,28,31,24,29,32,36,10,18,26,25,11,7,17,19);(1,2,3,4)(5,7)(6,11)(8,18)(9,25)(10,20)(12,29)(13,21)(14,26)(15,24)(16,23)(17,19)(22,36)(27,31)(28,30)(32,34)(33,35);(5,8,12,23,13,24,10,11)(6,7,18,29,16,21,15,20)(9,22,27,33,28,32,26,17)(14,19,25,36,31,35,30,34)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=ei,d^2=gh,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,f^b=fg,g^b=gh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72216258, #72216270, #72216271, … |
| maximal normal subgroup of | #72216258, #72216270, #72216271, … |
| maximal quotient of | #72225471 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60190);