Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C42 |
| commutator | C4 | C8×C16 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle, prime radical | C23 | #346 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2)(3,4)(13,14,16,22,17,24,28,37,18,20,29,32,30,33,41,42,19,23,27,36,31,38,40,44,15,21,25,34,26,35,39,43);(2,4)(5,6,7,9,8,10,11,12)(14,20,23,21)(16,27)(22,34,36,32)(24,33,38,35)(25,29)(28,40)(37,43,44,42)(39,41)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bc,e^a=ei,d^b=di176 conjugacy classes — view the full character table.
| maximal subgroup of | #72234237, #72286359, #72286363, … |
| maximal normal subgroup of | #72234237, #72286359, #72286363, … |
| maximal quotient of | #72234282, #72234299, #72234311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1186);