Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1062 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)(5,6,12,18)(7,14)(8,13)(9,15,20,19)(10,29,21,17)(11,30,28,16)(22,34)(23,33)(24,36)(25,32)(26,35)(27,31);(2,4)(5,7,20,25,12,26,9,22)(6,13,15,31,18,33,19,36)(8,21,24,28,23,10,27,11)(14,16,34,17,35,30,32,29);(5,8)(6,14)(7,21)(9,27)(10,26)(11,22)(12,23)(13,16)(15,34)(17,31)(18,35)(19,32)(20,24)(25,28)(29,36)(30,33);(5,9,12,20)(6,15,18,19)(7,22,26,25)(8,27,23,24)(10,28,21,11)(13,31,33,36)(14,34,35,32)(16,17,30,29)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=i,e^2=h,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=ehi,f^c=fh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239492, #72249884, #72249887, … |
| maximal normal subgroup of | #72239492, #72249884, #72249887, … |
| maximal quotient of | #72265681, #72285673, #72285735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401530);