Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23996 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,10,14,16)(8,11,15,18)(9,12)(13,19)(17,20);(1,3,4,2)(5,7)(6,10,12,16)(8,13)(9,14)(11,17,18,19)(15,20);(10,16)(17,19);(5,8)(6,11,12,18)(7,13)(9,15)(10,17,16,19)(14,20)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=g,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72225620, #72225639, #72225642, … |
| maximal normal subgroup of | #72225620, #72225639, #72225642, … |
| maximal quotient of | #72251461, #72265807, #72265808, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 278216);