Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533105 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3232 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)(21,22)(23,31)(24,29)(25,28)(26,30)(27,32)(33,35)(34,36);(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,23)(22,27)(24,32)(25,34)(26,33)(28,31)(29,35)(30,36);(2,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(21,24,26,25)(22,28,30,29)(23,32,33,34)(27,31,36,35);(8,16)(11,18)(14,20)(17,19)(21,24,26,25)(22,29,30,28)(23,32,33,34)(27,35,36,31)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,f^b=fi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72284013, #72284278 |
| maximal normal subgroup of | #72284013, #72284278 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6478822);