Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C24 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(5,6,9,7,8,10,12,11)(13,14)(15,21)(16,24)(17,20)(18,23)(19,25)(22,27)(26,28);(6,10)(7,11)(13,15)(14,18,20,26)(16,23)(17,22)(19,21,24,27)(25,28);(5,7,9,10,8,11,12,6)(15,22)(18,26)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ci,d^a=dh,f^a=fh,c^b=chi152 conjugacy classes — view the full character table.
| maximal subgroup of | #72212990, #72215908, #72215911, … |
| maximal normal subgroup of | #72212990, #72215908, #72215911, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12569);