Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30496 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6,10,15,12,7,11,16)(8,17,18,13,14,19,20,9);(5,7)(6,12)(8,17)(9,18)(10,15)(11,16)(13,20)(14,19);(5,8)(6,13)(7,9)(10,18)(11,20)(12,14)(15,19)(16,17);(5,9)(6,14)(7,8)(10,19)(11,17)(12,13)(15,18)(16,20)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265640, #72265644, #72265688, … |
| maximal normal subgroup of | #72265640, #72265644, #72265688, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290263);