Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2082 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)(7,10)(8,13)(9,11)(12,14)(15,20)(16,19)(17,18);(5,7,14,16,8,15,9,17)(6,10,11,18,13,20,12,19);(6,11)(7,15)(9,14)(10,18)(12,13)(19,20);(6,12)(9,14)(10,19)(11,13)(16,17)(18,20)Polycyclic presentation:
2^9|a^2=h,b^2=ef,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251691, #72251692, #72251713, … |
| maximal normal subgroup of | #72251691, #72251692, #72251713, … |
| maximal quotient of | #72237077 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291097);