Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420461 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,17,9,13,19,14)(7,12,15,11,16,10,18,20);(1,2)(3,4)(5,7)(6,10)(8,18)(9,16)(11,17)(12,13)(14,20)(15,19);(2,4)(5,7)(6,11,13,20)(8,15)(9,16)(10,14,12,17)(18,19);(5,7)(6,12,13,10)(8,18)(9,16)(11,17,20,14)(15,19)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72239534, #72252462, #72252466, … |
| maximal normal subgroup of | #72239534, #72252462, #72252466, … |
| maximal quotient of | #72239546, #72260913, C2×#420485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327681);