Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #121 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #386 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,9,11,8,7,10,12)(14,19)(15,22)(16,17)(20,27)(21,25)(23,24);(1,2,3,4)(6,7)(11,12)(13,14)(15,20)(16,25)(17,19)(18,21)(22,26)(23,28)(24,27);(5,7,9,12,8,6,10,11)(13,15,16,23,18,24,17,22)(14,20,25,28,21,27,19,26)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efh,d^2=h,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=ci,d^b=dh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72215865, #72215908, #72215911 |
| maximal normal subgroup of | #72215865, #72215908, #72215911 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17532);