Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3458 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)(9,10)(11,12)(13,14)(15,18)(16,20)(17,19);(5,6)(7,8)(13,14,15,20,17,19,16,18);(2,4)(6,8)(13,15,17,16)(14,18,19,20);(10,12)(13,14)(15,18)(16,20)(17,19)Polycyclic presentation:
2^9|b^2=ei,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=dei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72315860, #72315893, #72315949, … |
| maximal normal subgroup of | #72315860, #72315893, #72315949, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324961);