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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2333 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)(15,22)(16,20)(17,19)(18,23)(21,25)(24,27)(26,28);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,16,23)(14,18,20,22)(17,24,25,28)(19,26,21,27);(2,4)(5,7,9,8)(6,11,12,10)(14,19)(17,25)(18,26)(20,21)(22,27)(24,28);(1,2)(3,4)(5,8,9,7)(6,10,12,11)(15,23)(18,22)(24,28)(26,27);(2,4)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^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 | #72256949, #72256985, #72256989, … |
| maximal normal subgroup of | #72256949, #72256985, #72256989, … |
| maximal quotient of | #72264626, #72311357, #72311569, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484812);