Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | C2×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1316 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)(5,6,12,19,9,17,10,30,14,21,11,31,13,20,29,18)(7,22,24,35,32,16,28,33,25,36,27,15,23,34,8,26);(1,2,3,4)(5,7,9,23,14,25,13,32)(6,15,20,26,21,35,17,33)(8,10,27,11,28,29,24,12)(16,19,36,18,34,31,22,30);(5,8)(6,16)(7,10)(9,27)(11,23)(12,32)(13,24)(14,28)(15,19)(17,22)(18,26)(20,36)(21,34)(25,29)(30,33)(31,35);(5,9,14,13)(6,17,21,20)(7,23,25,32)(8,27,28,24)(10,11,29,12)(15,33,35,26)(16,22,34,36)(18,19,30,31)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,e^b=eh,f^b=fhi,g^b=gh,h^b=hi,e^c=ehi,f^c=fh,g^c=ghi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249881, #72249890, #72249903, … |
| maximal normal subgroup of | #72249881, #72249890, #72249903, … |
| maximal quotient of | #72265643, #72285706, #72285779, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401761);