Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460096 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11)(8,12)(9,20)(10,14)(13,22)(15,23)(16,32)(17,25)(18,27)(19,28)(21,29)(24,34)(26,31)(30,35)(33,36);(5,7,10,17)(6,11,14,25)(8,15,19,31)(9,16,22,34)(12,26,28,23)(13,24,20,32)(18,30,21,33)(27,36,29,35);(1,2,3,4)(5,8)(6,12)(7,15)(9,21,22,18)(10,19)(11,23)(13,29,20,27)(14,28)(16,33,34,30)(17,31)(24,36,32,35)(25,26);(5,9)(6,13,14,20)(7,16)(8,18)(10,22)(11,24,25,32)(12,27,28,29)(15,30)(17,34)(19,21)(23,35,26,36)(31,33)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=g,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^c=de,e^c=ei,h^c=hi,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72284233, #72284470, #72284660, … |
| maximal normal subgroup of | #72284233, #72284470, #72284660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376875);