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 | #2397 |
| prime radical | C2×#2017 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,12,16,11,28)(7,17,20,31,21,33,22,13)(8,14,24,9,25,26,29,27)(18,32,23,30,35,34,36,19);(5,7)(6,13)(8,23)(9,19)(10,22)(11,20)(12,21)(14,32)(15,33)(16,31)(17,28)(18,24)(25,36)(26,34)(27,30)(29,35);(5,8)(6,14)(7,18)(9,15)(10,24)(11,29)(12,25)(13,30)(16,26)(17,34)(19,31)(20,23)(21,35)(22,36)(27,28)(32,33);(2,4)(5,9,10,26,12,27,11,14)(6,8,15,24,16,25,28,29)(7,19,22,34,21,30,20,32)(13,23,33,18,31,36,17,35)Polycyclic presentation:
2^9|a^2=e,d^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 | #72250134, #72257164, #72257191, … |
| maximal normal subgroup of | #72250134, #72257164, #72257191, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307309);