Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| 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×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
903 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)(6,10)(8,15)(11,17)(12,16)(14,19)(18,20)(21,22,26,24,25,28,33,32)(23,27,31,30,29,34,36,35);(5,6,9,12)(7,11,13,18)(8,16,15,10)(14,20,19,17)(21,23)(22,27)(24,30)(25,29)(26,31)(28,34)(32,35)(33,36);(2,4)(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,19)(16,20)(22,28)(23,29)(24,32)(31,36);(6,12)(7,13)(10,16)(14,19)(21,24,26,28,25,32,33,22)(23,30,31,34,29,35,36,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ci,d^b=dh,e^b=eh,d^c=dhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116576);