Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456658 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| 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 | |
1141 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)(8,20)(9,14)(10,13)(12,23)(15,24)(16,30)(17,27)(18,28)(19,29)(21,35)(22,31)(25,32)(26,33)(34,36);(5,7)(6,11,14,29)(8,21)(9,19)(10,17)(12,30,32,36)(13,27,15,28)(16,23)(18,24)(20,35,31,26)(22,33)(25,34);(2,4)(5,8,9,22)(6,12,14,32)(7,16,19,34)(10,25,24,23)(11,26,29,35)(13,20,15,31)(17,21,18,33)(27,30,28,36);(6,13)(8,22)(10,24)(11,27)(12,31)(14,15)(17,18)(20,32)(21,33)(26,30)(28,29)(35,36)Polycyclic presentation:
2^9|b^2=egh,c^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285151, #72285470, #72285522, … |
| maximal normal subgroup of | #72285151, #72285470, #72285522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378642);