Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994107 |
| 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 | C2×#3217 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,31)(24,29)(25,35)(26,28)(27,33)(30,32)(34,36);(7,13)(8,16)(10,17)(11,15)(21,23,32,34,26,24,25,33)(22,27,35,29,28,36,30,31);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(22,28)(23,24)(30,35)(33,34);(7,13)(10,17)(14,19)(18,20)(21,24,26,23)(22,29,28,31)(25,33,32,34)(27,30,36,35);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,28)(27,36)(29,31)(30,35)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72282925, #72283184, #72283589, … |
| maximal normal subgroup of | #72282925, #72283184, #72283589, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7074206);