Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19671 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#390 |
| prime radical | C2×#3169 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,13,16,14,17,12,22)(11,18,21,23,19,24,20,15);(2,6)(5,8)(9,11)(10,15,17,23)(12,20)(13,21)(14,19)(16,18,22,24);(1,2,3,5,4,6,7,8);(11,19)(15,23)(18,24)(20,21)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ci,d^b=dh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72286365, #72286405, #72286454, … |
| maximal normal subgroup of | #72286365, #72286405, #72286454, … |
| maximal quotient of | #72321915 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 113564);