Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| 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×#3224 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1888 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)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22)(23,29)(24,31)(25,30)(26,32)(27,33)(28,34)(35,36);(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,26,34)(22,27,32,36)(24,33,25,35)(28,30,29,31);(1,2,3,4)(6,11)(7,13)(12,15)(14,19)(21,23)(22,28)(24,35)(25,33)(26,34)(27,31)(29,32)(30,36);(7,13)(10,17)(14,19)(18,20)(21,23)(22,29)(24,35)(25,33)(26,34)(27,30)(28,32)(31,36);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(21,24,26,25)(22,30,32,31)(23,33,34,35)(27,29,36,28)Polycyclic presentation:
2^9|b^2=i,c^2=gh,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405674);