Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#383 |
| prime radical | C25 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1774 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)(7,16)(8,21,24,35)(9,25)(10,14)(11,15)(12,19)(13,23,28,22)(17,31,32,26)(18,30)(20,27)(29,34,36,33);(5,7)(6,12,15,27)(8,17)(9,19)(10,20)(11,18)(13,26,28,31)(14,16,25,30)(21,29,35,36)(22,33)(23,34)(24,32);(2,4)(5,8,9,22,11,24,10,23)(6,13,14,21,15,28,25,35)(7,17,19,33,18,32,20,34)(12,26,16,29,27,31,30,36);(7,18)(12,27)(16,30)(17,32)(19,20)(26,31)(29,36)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=di,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72250758, #72250767, #72250791, … |
| maximal normal subgroup of | #72250758, #72250767, #72250791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260714);