Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×D4×#41 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,24)(16,21)(17,22)(18,27)(19,23)(20,25)(26,28);(1,2,3,4)(6,9)(10,11)(13,15,18,25,19,16,17,26)(14,20,22,24,23,28,27,21);(5,7,8,12)(6,10,9,11)(13,16)(14,21)(15,19)(17,26)(18,25)(20,27)(22,28)(23,24);(13,16)(14,21)(15,19)(17,26)(18,25)(20,27)(22,28)(23,24);(13,17,19,18)(14,22,23,27)(15,25,16,26)(20,21,28,24)Polycyclic presentation:
2^9|b^2=fg,c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72257732, #72308658, #72308957, … |
| maximal normal subgroup of | #72257732, #72308658, #72308957, … |
| maximal quotient of | #72257945, #72315917 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353884);