Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #102 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C22×C4×C8 | C22 |
| socle | C23 | D4⋊C8 |
| prime radical | C24 | #104 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,18)(16,25)(17,21)(19,23)(20,22)(24,27)(26,28);(5,7)(6,9,10,11)(8,12)(14,18,20,26,21,27,25,19)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,e^2=h,f^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,h^a=hi,c^b=cg116 conjugacy classes — view the full character table.
| maximal subgroup of | #72210223, #72210317, C2×#93386, … |
| maximal normal subgroup of | #72210317, C2×#93386, #72213037, … |
| maximal quotient of | #72210220, #72210666, #72210667, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 582);