Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2079 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(13,14,18,20,17,22,16,21)(15,19,23,28,24,27,25,26);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,25)(17,24)(18,23)(20,28)(21,26)(22,27);(1,3,4,2)(6,9)(10,12)(13,14,17,22)(15,19,24,27)(16,20,18,21)(23,26,25,28);(5,7)(6,10)(8,11)(9,12)(13,16,17,18)(14,20,22,21)(15,23,24,25)(19,26,27,28);(7,11)(10,12)(15,24)(19,27)(23,25)(26,28)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ei,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
| maximal subgroup of | #72282978, #72282991, #72284610, … |
| maximal normal subgroup of | #72282978, #72282991, #72284610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6459859);