Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| 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 | #2366 |
| prime radical | #23421 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1469 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)(5,6,10,17,12,18,11,16)(7,19,22,31,23,35,21,13)(8,24,26,32,27,36,25,14)(9,20,29,34,30,33,28,15);(5,7)(6,13)(8,15)(9,14)(10,22)(11,21)(12,23)(16,35)(17,19)(18,31)(20,26)(24,29)(25,33)(27,34)(28,36)(30,32);(1,2,3,4)(5,8,12,27)(6,14,18,32)(7,20,23,33)(9,19,30,35)(10,26,11,25)(13,29,31,28)(15,22,34,21)(16,36,17,24);(5,9,10,29,12,30,11,28)(6,15,17,20,18,34,16,33)(7,14,22,24,23,32,21,36)(8,19,26,31,27,35,25,13)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=h,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,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 284175);