Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134415 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1804 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,17,11,14,10,16)(7,18,19,31,22,12,21,30)(8,23,26,33,27,13,25,15)(20,35,34,28,24,32,36,29);(1,2)(3,4)(5,7,10,19,11,22,9,21)(6,12,16,30,14,18,17,31)(8,24,25,36,27,20,26,34)(13,32,33,29,23,35,15,28);(2,4)(6,13,14,23)(7,19)(8,25)(9,10)(12,28,18,29)(15,17,33,16)(20,24)(21,22)(26,27)(30,35,31,32)Polycyclic presentation:
2^9|a^2=g,b^2=dh,c^2=d,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,e^c=eh,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249862, #72249878, #72249885, … |
| maximal normal subgroup of | #72249862, #72249878, #72249885, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55940);