Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394823 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2416 |
| 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 | |
1208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,11)(8,16)(9,22)(10,18)(12,15)(13,20)(14,24)(17,25)(19,27)(21,28)(23,30)(26,29)(31,33)(32,35)(34,36);(5,7,11,6,12,18,10,15)(8,13,26,28,21,29,20,16)(9,17,24,14,25,22,23,30)(19,27,33,35,34,36,32,31);(1,2,4,3)(5,8,10,20,12,21,11,26)(6,13,7,16,15,29,18,28)(9,19,23,32,25,34,24,33)(14,27,17,31,30,36,22,35);(1,3,4,2)(5,9,12,25)(6,14,15,30)(7,17,18,22)(8,19,21,34)(10,23,11,24)(13,27,29,36)(16,31,28,35)(20,32,26,33)Polycyclic presentation:
2^9|b^2=ei,c^2=f,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308482);