Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #164512 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1035 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11)(8,16,18,12)(9,13)(10,14)(15,20,17,19)(21,22)(23,29,31,36)(24,28)(25,27)(26,30,34,32)(33,35);(1,2)(3,4)(5,7,10,9)(6,11,14,13)(8,15,18,17)(12,19,16,20);(5,8)(6,12)(7,15)(9,17)(10,18)(11,19)(13,20)(14,16)(21,23)(22,26)(24,32)(25,31)(27,34)(28,36)(29,35)(30,33);(2,4)(5,9,10,7)(6,13,14,11)(8,17,18,15)(12,20,16,19)(22,27)(26,34)(28,35)(29,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=de,f^c=fi,g^c=ghi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 128941);