Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455822 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| 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 | |
1109 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)(6,11)(8,19)(10,12)(13,23)(14,28)(16,32)(18,21)(20,22)(24,30)(26,35)(29,36)(33,34);(5,6,12,25,9,11,10,15)(7,13,21,31,17,23,18,27)(8,14,19,30,20,24,22,28)(16,26,32,36,33,29,34,35);(2,4)(5,7,9,17)(6,13,11,23)(8,16,20,33)(10,21,12,18)(14,29,24,26)(15,31,25,27)(19,34,22,32)(28,35,30,36);(5,8)(6,14)(7,16)(9,20)(10,22)(11,24)(12,19)(13,26)(15,28)(17,33)(18,34)(21,32)(23,29)(25,30)(27,35)(31,36)Polycyclic presentation:
2^9|b^2=egh,c^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,h^c=hi,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378631);