Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393989 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| prime radical | #34632 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,19)(12,16)(13,25)(14,18)(15,21)(17,27)(20,35)(22,33)(23,31)(24,39)(26,29)(28,40)(30,37)(32,38)(34,36);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,20)(13,26)(14,22)(16,30)(17,33)(18,29)(19,25)(21,27)(23,36)(24,40)(28,31)(32,34)(35,39)(37,38);(9,12)(10,16)(11,20)(13,24)(14,23)(15,28)(17,32)(18,31)(19,34)(21,37)(22,36)(25,39)(26,40)(27,38)(29,30)(33,35);(2,6)(5,8)(12,23)(16,31)(20,36)(24,38)(28,30)(32,39)(34,35)(37,40)Polycyclic presentation:
2^9|b^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72237199, #72257064, #72257087, … |
| maximal normal subgroup of | #72237199, #72257064, #72257087, … |
| maximal quotient of | #72237249, #72237253, #72238730, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312418);