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) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C42 |
| commutator | C4 | C8×C16 |
| Frattini, cosocle | C2×C82 | C22 |
| socle, prime radical | C23 | #322 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(3,4)(5,6,8,13,9,11,17,18,10,14,16,20,7,12,15,19);(2,4)(6,11,14,12)(8,16)(13,19,20,18)(15,17)(21,22,23,26,24,27,29,32,25,28,30,33,31,34,35,36)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=h,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bc,e^a=eh,d^b=dh176 conjugacy classes — view the full character table.
| maximal subgroup of | #72234227, #72286315, #72287022, … |
| maximal normal subgroup of | #72234227, #72286315, #72287022, … |
| maximal quotient of | #72210714, #72234264, #72234299, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92);