Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #612424 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×#121 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
7846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(10,12)(14,16)(15,19);(1,2,4,3)(6,8)(10,12)(13,14,17,16)(15,18,19,20);(5,6)(7,8)(9,10)(11,12)(13,14,17,16)(15,18,19,20);(13,15)(14,18)(16,20)(17,19);(1,3,4,2)(13,16,17,14)(15,20,19,18)Polycyclic presentation:
2^9|b^2=f,c^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch116 conjugacy classes — view the full character table.
| maximal subgroup of | #72237107, #72237110, #72237112, … |
| maximal normal subgroup of | #72237107, #72237110, #72237112, … |
| maximal quotient of | #72307003, #72334312, C2×#615501, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 522697);