Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(9,10,11,12)(13,14,18,20)(15,16,19,17);(5,6)(7,8)(14,15)(16,17)(19,20);(2,4)(6,8)(13,15,17,14,18,19,16,20);(13,16,18,17)(14,19,20,15)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^b=ei,f^b=fi,g^b=gi,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72313102, #72316862, #72316869, … |
| maximal normal subgroup of | #72313102, #72316862, #72316869, … |
| maximal quotient of | #72313230, #72317054 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265152);