Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393988 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2692 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)(7,17)(8,21)(9,15)(10,14)(11,16)(12,19)(13,22)(18,26)(20,30)(23,31)(24,27)(25,29)(28,35)(32,34)(33,36);(5,7,9,19,11,20,10,24)(6,12,14,17,16,27,15,30)(8,18,22,33,23,35,25,34)(13,28,21,36,29,26,31,32);(5,8)(6,13)(7,18)(9,22)(10,25)(11,23)(12,26)(14,21)(15,31)(16,29)(17,32)(19,33)(20,35)(24,34)(27,28)(30,36);(2,4)Polycyclic presentation:
2^9|b^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250092, #72256881, #72256923, … |
| maximal normal subgroup of | #72250092, #72256881, #72256923, … |
| maximal quotient of | C2×#394197 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301393);