Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
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 | #60643 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,18,11,17,12,16)(7,19,21,13,22,29,8,14)(9,24,25,15,26,32,27,31)(20,30,34,33,35,28,23,36);(1,2,4,3)(5,7,11,22)(6,13)(8,10,21,12)(9,20,26,35)(14,17)(15,28)(16,19)(18,29)(23,25,34,27)(24,33)(30,31)(32,36);(5,8,10,7,11,21,12,22)(6,14,16,29,17,13,18,19)(9,23,25,20,26,34,27,35)(15,30,24,36,31,28,32,33);(1,3,4,2)(5,9,10,25,11,26,12,27)(6,15,16,24,17,31,18,32)(7,20,21,34,22,35,8,23)(13,28,19,33,14,30,29,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fg,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276401);