Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #117 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C22 | #2163 |
| prime radical | C23 | C2×M32 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
234 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 146 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)(7,16)(12,15)(14,23)(17,30)(18,24)(26,29)(31,34);(2,4)(5,6,7,13,8,14,17,28,9,15,18,27,19,29,31,36,10,11,16,22,20,23,30,33,21,12,24,25,32,26,34,35);(6,12,11,15)(7,16)(13,27,22,25)(14,26,23,29)(17,30)(18,24)(28,36,33,35)(31,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,f^2=g,g^2=h,h^2=i,b^a=bd,f^a=fi,c^b=ch,d^b=di,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216243, #72216247, #72216248, … |
| maximal normal subgroup of | #72216243, #72216247, #72216248, … |
| maximal quotient of | #72225407, #72225413, #72227193 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59177);