Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C22×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,20)(17,19)(18,22)(23,27)(24,26)(25,28);(7,11)(10,12)(13,15)(14,18)(16,23)(17,22)(19,21)(20,25)(24,28)(26,27);(6,9)(10,12)(15,22)(18,21)(23,28)(25,27);(5,7,8,11)(6,10,9,12)(14,19)(18,21)(20,26)(25,27);(6,9)(10,12);(13,16,17,24)(14,20,19,26)(15,23,22,28)(18,25,21,27)Polycyclic presentation:
2^9|a^2=i,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=cg,d^b=dh152 conjugacy classes — view the full character table.
| maximal subgroup of | #72283835, #72304540, #72304570, … |
| maximal normal subgroup of | #72283835, #72304540, #72304570, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533162);