Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | #60422 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1715 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,16)(8,21,25,32)(9,14)(10,15)(11,18)(12,24,30,23)(13,26)(17,27,34,36)(19,28)(20,31)(22,29,35,33);(5,7)(6,11)(8,22,25,35)(9,19)(10,20)(12,29,30,33)(13,31)(14,16)(15,28)(17,24,34,23)(18,26)(21,36,32,27);(1,2,3,4)(5,8,10,25)(6,12,15,30)(7,17,20,34)(9,23,26,24)(11,27,28,36)(13,21,14,32)(16,33,31,29)(18,22,19,35);(5,7)(6,11)(8,17)(9,19)(10,20)(12,27)(13,31)(14,16)(15,28)(18,26)(21,29)(22,24)(23,35)(25,34)(30,36)(32,33)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276042);