Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24218 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,23)(16,21)(17,22)(18,27)(19,25)(20,24)(26,28);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,19)(16,26)(17,25)(18,24)(20,21)(22,23)(27,28);(2,4)(13,15)(14,20)(16,24)(17,25)(18,26)(19,21)(22,28)(23,27);(1,2)(3,4)(5,7,9,8)(6,11,12,10)(13,16,17,18)(14,21,22,27)(15,24,25,26)(19,28,23,20);(2,4)(5,8,9,7)(6,10,12,11)Polycyclic presentation:
2^9|b^2=fi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
| maximal subgroup of | #72257477, #72257520, #72263928, … |
| maximal normal subgroup of | #72257477, #72257520, #72263928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485983);