Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458129 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| 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 | |
891 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)(6,12)(7,17)(8,22)(9,10)(13,27)(15,16)(19,35)(20,21)(24,25)(26,31)(28,30)(34,36);(5,6,9,12,11,16,10,15)(7,18,17,27,21,29,20,13)(8,14,24,26,25,32,22,31)(19,34,33,36,35,30,23,28);(5,7,9,17,11,21,10,20)(6,13,12,18,16,27,15,29)(8,23,24,19,25,33,22,35)(14,30,26,28,32,34,31,36);(2,4)(5,8,9,24,11,25,10,22)(6,14,12,26,16,32,15,31)(7,19,20,23,21,35,17,33)(13,28,29,30,27,36,18,34)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=e,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,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365325);