Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470122 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,24)(16,22)(17,23)(18,21)(19,26)(20,25)(27,28);(2,4)(7,8)(10,11)(13,15)(14,19)(16,25)(17,26)(18,27)(20,21)(22,24)(23,28);(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,25)(17,26)(18,27)(19,21)(22,28)(23,24);(5,8,9,7)(6,11,12,10)(14,21)(19,20)(22,23)(24,28);(6,12)(10,11)(13,16,18,17)(14,22,21,23)(15,25,27,26)(19,24,20,28)Polycyclic presentation:
2^9|c^2=h,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257732, #72315662, #72340631, … |
| maximal normal subgroup of | #72257732, #72315662, #72340631, … |
| maximal quotient of | #72257947, #72257951, #72316029 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394751);