Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,7)(13,14,17,20)(15,18,16,19);(2,4)(9,10,11,12)(14,18)(15,16)(19,20);(14,19)(15,16)(18,20);(5,7,8,6)(14,20)(18,19)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=gi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,e^b=ei,f^b=fi,e^c=ei,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72237131, #72237154, #72239112, … |
| maximal normal subgroup of | #72237131, #72237154, #72239112, … |
| maximal quotient of | #72226850, #72242475, #72249880, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261453);