Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500401 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,25,28)(23,29,30,36)(24,32,33,27)(26,34,35,31);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,24,31,25,30,33,34)(22,26,27,29,28,35,32,36);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(23,30)(26,35)(29,36)(31,34);(8,16)(11,20)(13,15)(17,19);(7,14)(10,18)(13,15)(17,19)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dhi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72314723, #72314731, #72314758, … |
| maximal normal subgroup of | #72314723, #72314731, #72314758, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6418015);