Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393899 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#399 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,14,10,15,11,25)(7,16,18,28,20,32,19,12)(8,21,23,31,24,13,26,30)(17,33,22,36,34,29,35,27);(5,7,11,18,10,20,9,19)(6,12,25,16,15,28,14,32)(8,22,26,34,24,35,23,17)(13,29,31,27,21,33,30,36);(5,8)(6,13)(7,17)(9,23)(10,24)(11,26)(12,27)(14,30)(15,21)(16,33)(18,22)(19,35)(20,34)(25,31)(28,36)(29,32);(2,4)Polycyclic presentation:
2^9|a^2=e,b^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72250020, #72257147, #72257191, … |
| maximal normal subgroup of | #72250020, #72257147, #72257191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303697);