Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,11)(7,15)(9,22)(12,27)(13,29)(14,26)(16,24)(17,32)(20,23)(25,31)(28,36)(30,34);(1,2)(3,4)(5,6,9,11,10,14,22,26)(7,16,33,24,15,12,18,27)(8,13,23,25,21,31,20,29)(17,34,19,36,32,28,35,30);(2,4)(5,7,9,18,10,15,22,33)(6,12,11,24,14,16,26,27)(8,19,20,32,21,35,23,17)(13,30,29,34,31,36,25,28);(5,8,10,21)(6,13,14,31)(7,17,15,32)(9,23,22,20)(11,25,26,29)(12,28,16,34)(18,35,33,19)(24,36,27,30)Polycyclic presentation:
2^9|b^2=ef,c^2=e,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285452, #72285522, #72285577, … |
| maximal normal subgroup of | #72285452, #72285522, #72285577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363155);