Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483839 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,11,17)(7,18)(8,20)(9,15,12,29)(10,25,27,30)(13,26)(14,33)(16,19,35,28)(21,22)(23,24)(31,36)(32,34);(1,2,3,4)(5,7,19,34,9,24,27,14,11,21,28,36,12,20,10,26)(6,13,30,23,29,31,35,18,17,33,25,8,15,32,16,22);(5,8,10,13,12,18,28,32,11,23,27,33,9,22,19,31)(6,14,16,20,15,34,25,21,17,26,35,24,29,36,30,7);(5,9,11,12)(6,15,17,29)(7,20,21,24)(8,22,23,18)(10,19,27,28)(13,31,33,32)(14,34,26,36)(16,25,35,30)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=i,e^2=h,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224826, #72224939, #72226810, … |
| maximal normal subgroup of | #72224826, #72224939, #72226810, … |
| maximal quotient of | #72224820, #72310165, #72310177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402438);