Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,11,17,25)(8,12,20,27)(9,14)(10,13)(15,18,26,23)(16,30)(19,22)(21,28,33,34)(24,32)(29,31)(35,36);(1,2)(3,4)(5,7)(6,11,13,25)(8,16)(9,18)(10,17)(12,24,27,30)(14,26,22,15)(19,23)(20,32)(21,31)(28,36,34,29)(33,35);(2,4)(5,6)(7,15)(8,12)(9,22)(10,13)(11,23)(14,19)(16,29)(17,26)(18,25)(20,27)(21,34)(24,35)(28,33)(30,31)(32,36);(5,8,19,33,10,20,9,21)(6,12,14,28,13,27,22,34)(7,16,23,35,17,32,18,31)(11,24,26,36,25,30,15,29)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=efgi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279976);