Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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 | #60450 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,16,11,17,30,15)(7,18,20,12,22,32,21,33)(8,23,24,34,26,31,25,13)(9,27,28,14,29,35,19,36);(1,2,3,4)(5,7)(6,12,17,33)(8,9)(10,20)(11,22)(13,27,34,35)(14,31,36,23)(15,18,16,32)(19,25)(21,30)(24,28)(26,29);(2,4)(5,8)(6,13)(7,19)(9,20)(10,24)(11,26)(12,14)(15,31)(16,23)(17,34)(18,27)(21,29)(22,28)(25,30)(32,35)(33,36);(5,9,10,28,11,29,30,19)(6,14,15,27,17,36,16,35)(7,8,20,24,22,26,21,25)(12,31,18,34,33,23,32,13)Polycyclic presentation:
2^9|a^2=h,b^2=efgh,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,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, 274303);