Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30056 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1234 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)(7,15)(8,11)(9,13)(10,14)(12,18)(16,19)(17,20);(5,7,9,8,10,17,18,16)(6,11,12,15,14,19,13,20);(7,16)(8,17)(9,18)(11,15)(12,13)(19,20);(5,8,18,7,10,16,9,17)(6,11,12,15,14,19,13,20)Polycyclic presentation:
2^9|a^2=h,b^2=e,d^2=gi,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224750, #72224754, #72224755, … |
| maximal normal subgroup of | #72224750, #72224754, #72224755, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288372);