Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456884 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1184 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,11,14,10,26)(7,16,18,28,19,29,25,12)(8,20,22,32,24,31,23,13)(17,27,34,30,35,36,21,33);(1,2)(3,4)(5,7,11,19)(6,12,14,28)(8,21)(9,25,10,18)(13,30)(15,29,26,16)(17,23)(20,27)(22,35)(24,34)(31,36)(32,33);(2,4)(5,8,9,23,11,24,10,22)(6,13,15,31,14,32,26,20)(7,17,18,21,19,35,25,34)(12,27,16,33,28,36,29,30);(6,14)(8,22)(9,10)(12,28)(13,31)(17,34)(18,25)(20,32)(21,35)(23,24)(27,33)(30,36)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72319988, #72320000, #72320049 |
| maximal normal subgroup of | #72319988, #72320000, #72320049 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364080);