Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#453 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2415 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(5,6)(7,11)(8,17)(9,18)(10,15)(12,16)(13,19)(14,20);(5,7,10,6,12,15,11,16)(8,18,20,13,9,17,19,14);(2,4)(5,8)(6,13)(7,17)(9,12)(10,19)(11,20)(14,16)(15,18);(5,9)(6,14)(7,17)(8,12)(10,19)(11,20)(13,16)(15,18)Polycyclic presentation:
2^9|a^2=h,b^2=fghi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72226293, #72241302, #72241304, … |
| maximal normal subgroup of | #72226293, #72241302, #72241304, … |
| maximal quotient of | #72226235, C2×#366736 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273932);