Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #379 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 36 points:
(6,10)(8,15)(11,19)(12,16)(13,17)(18,20)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36);(1,2,3,4)(5,6)(7,11)(8,16)(9,12)(10,15)(13,18)(14,20)(17,19)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(5,7,9,14)(6,11,12,20)(8,17,15,13)(10,18,16,19)(23,29)(26,32)(30,35)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=ci,d^b=dh,e^b=eh,d^c=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 13555);