Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C82 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4×C8 | C42 |
| commutator | C4 | C2×C82 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C22 | C2×#346 |
| prime radical | C23 | #346 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,10,11,17,12,18,21,32,13,19,22,31,23,33,35,40,14,16,20,29,24,30,34,39,25,15,26,27,36,28,37,38);(1,2,3,5,4,6,7,8)(10,15,16,19)(11,20)(17,31,29,27)(18,28,30,33)(21,34)(22,26)(32,40,39,38)(35,37);(10,16)(15,19)(17,29)(18,30)(27,31)(28,33)(32,39)(38,40)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bh,c^a=ci,e^a=ei,d^b=di176 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#1398, #72216311, #72216324, … |
| maximal normal subgroup of | C8×#1398, #72216311, #72216324, … |
| maximal quotient of | #72216226, C2×#149495, #72286359 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56691);