Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12933 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1853 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6,10,17,12,18,11,28)(7,19,22,33,24,35,23,29)(8,16,26,14,27,32,9,15)(13,25,30,20,31,21,34,36);(5,7)(6,13,18,31)(8,25)(9,20)(10,23)(11,22)(12,24)(14,29,15,33)(16,19,32,35)(17,34,28,30)(21,27)(26,36);(1,3,4,2)(6,14,18,15)(8,9)(10,11)(13,29,31,33)(16,28,32,17)(19,30,35,34)(20,25)(21,36)(22,23)(26,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dh,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240833, #72249880, #72249881, … |
| maximal normal subgroup of | #72240833, #72249880, #72249881, … |
| maximal quotient of | #72240835, #72279935, C2×#147287 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54483);