Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#466 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,11)(8,9)(12,18)(13,17)(14,15)(19,20)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36);(5,6,10,13)(7,12,16,19)(8,11,9,17)(14,18,15,20)(23,29)(26,32)(30,35)(33,36);(2,4)(5,7,10,16)(6,12,13,19)(8,14,9,15)(11,18,17,20)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(5,8)(7,14)(9,10)(11,17)(15,16)(18,20)(23,29)(26,32)(30,35)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,c^b=ci,d^b=de,e^b=eh,d^c=di,e^d=eh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 127357);