Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61109 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(6,11)(7,13)(9,19)(14,15)(16,20)(17,18);(5,6)(7,12)(8,16)(9,15)(10,14)(11,19)(13,17)(18,20);(5,7,9,13,10,16,19,20)(6,12,15,17,14,8,11,18);(5,8)(6,13)(7,11)(9,18)(10,12)(14,20)(15,16)(17,19);(6,14)(7,16)(11,15)(13,20)Polycyclic presentation:
2^9|a^2=h,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,d^b=df,e^b=ei,f^b=fi,e^c=ei,f^d=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72221485, C4×#61110, #72258306, … |
| maximal normal subgroup of | C4×#61110, #72258306, #72258315 |
| maximal quotient of | #72221485 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287374);