Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#423 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,15,18,30)(8,12)(9,14)(10,13)(11,17,25,22)(16,26,31,35)(19,32,33,24)(20,28)(21,27)(23,29)(34,36);(5,7,9,22,10,18,23,17)(6,11,14,30,13,25,29,15)(8,19,20,16,21,33,34,31)(12,26,28,24,27,35,36,32);(1,2,3,4);(5,8,9,20,10,21,23,34)(6,12,14,28,13,27,29,36)(7,16,17,19,18,31,22,33)(11,24,15,26,25,32,30,35);(6,13)(11,25)(12,27)(14,29)(15,30)(24,32)(26,35)(28,36)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,d^2=f,f^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=df,f^b=fi,h^b=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72313252, #72313257, #72313312, … |
| maximal normal subgroup of | #72313252, #72313257, #72313312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289939);