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 | |
913 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)(5,6,10,14)(7,15,8,12)(9,18,19,13)(11,17,20,16)(21,22,26,24,25,28,33,32)(23,27,31,30,29,34,36,35);(5,7,10,8)(6,11,14,20)(9,17,19,16)(12,18,15,13)(21,23)(22,27)(24,30)(25,29)(26,31)(28,34)(32,35)(33,36);(2,4)(5,8,10,7)(6,12,14,15)(9,17,19,16)(11,18,20,13)(22,28)(23,29)(24,32)(31,36);(7,8)(11,20)(12,15)(16,17)(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,c^2=h,d^2=gh,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, 116579);