Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C25 | C24 |
| commutator | C24 | C23×C4 |
| prime radical | C22×#392 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
6918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,8,7)(10,11)(14,15);(9,10,12,11)(18,20);(13,14,16,15)(17,18)(19,20);(1,2,3,4)(5,7,8,6)(9,11,12,10)(13,15,16,14)Polycyclic presentation:
2^9|a^2=i,b^2=e,c^2=f,d^2=hi,b^a=be,c^a=cf,d^a=dh,c^b=cg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72278166, #72278240, #72278261, … |
| maximal normal subgroup of | #72278166, #72278240, #72278261, … |
| maximal quotient of | #72299702, #72299902, C2×#154647 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 61843);