Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500829 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #25779 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
879 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(14,20)(15,25)(16,27)(17,28)(18,21)(22,23)(24,26);(1,2)(3,4)(6,9)(10,12)(13,14,17,22,18,23,16,20,19,24,27,15,21,25,28,26);(2,4)(5,7,8,11)(6,10,9,12)(13,15,16,14,21,20,17,25,19,22,28,24,18,26,27,23);(6,9)(10,12)(13,14,17,22,18,23,16,20,19,24,27,15,21,25,28,26)Polycyclic presentation:
2^9|b^2=f,c^2=e,d^2=f,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg98 conjugacy classes — view the full character table.
| maximal subgroup of | #72318484, #72319524, #72319542, … |
| maximal normal subgroup of | #72318484, #72319524, #72319542, … |
| maximal quotient of | #72322637, #72322655 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 412390);