Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×C42 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | C2×#346 |
| prime radical | C23 | C2×M32 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
236 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 150 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)(6,13,15,18)(9,24)(14,28,17,19)(16,33,32,29)(21,25)(23,27)(26,30)(31,35,34,36);(2,4)(5,6,9,16,10,17,26,34,11,18,27,33,22,28,25,36,12,15,24,32,20,14,30,31,8,13,23,29,7,19,21,35);(5,7,8,20,12,22,11,10)(6,14,18,19,15,17,13,28)(9,25,23,26,24,21,27,30)(16,34,33,36,32,31,29,35)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,f^a=fi,c^b=ch,e^b=ei,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72225413, #72225508, #72225521, … |
| maximal normal subgroup of | #72225413, #72225508, #72225521, … |
| maximal quotient of | #72225407, #72225413, #72227193 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59175);