Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499724 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14,17,22,21,16,28,27,20,24,18,23,19,15,25,26);(1,2)(3,4)(6,9)(10,11)(13,14)(15,21)(16,19)(17,26)(18,27)(20,24)(22,25)(23,28);(2,4)(13,15,20,16)(14,21,24,19)(17,23,18,22)(25,26,28,27);(5,7,8,12)(6,10,9,11)(13,16,20,15)(14,19,24,21)(17,22,18,23)(25,27,28,26)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=efh,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi,e^d=eh,f^d=fh,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318893, #72319542, #72319600, … |
| maximal normal subgroup of | #72318893, #72319542, #72319600, … |
| maximal quotient of | #72322595 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411968);