Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150138 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C22×#508 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
645 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 645 conjugacy classes of subgroups — too many to draw. Only its 108 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 — 3 generators on 36 points:
(1,2)(3,4)(6,11,12,27)(7,15)(8,20)(9,22)(13,31,30,35)(14,32,24,29)(16,21,28,26)(17,23)(18,36)(19,25)(33,34);(5,6)(7,16)(8,21)(9,24)(10,12)(11,25)(13,20)(14,22)(15,27)(17,30)(18,31)(19,28)(23,26)(29,34)(32,36)(33,35);(2,4)Polycyclic presentation:
2^9|a^2=f,d^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dgi,f^b=fi,g^b=gh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72286674, #72319634, D4×#1418, … |
| maximal normal subgroup of | #72286674, #72319634, D4×#1418, … |
| maximal quotient of | #72286707, #72321070 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59357);