Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | C2×#3025 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2105 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)(7,13,15,19)(8,11)(9,12)(10,16,14,17)(18,20);(5,6)(7,14,15,10)(8,11)(9,12)(13,16,19,17)(18,20);(5,7)(6,10)(8,17)(9,15)(11,19)(12,14)(13,20)(16,18);(2,4)(5,6)(7,10)(8,11)(9,12)(13,16)(14,15)(17,19)(18,20)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72233138, #72237986 |
| maximal normal subgroup of | #72233138, #72237986 |
| maximal quotient of | #72233133, #72236996, #72237976 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325164);