Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
908 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(6,10)(7,14)(8,11)(12,15)(13,18)(16,19)(21,22,24,27,26,30,25,29)(23,31,33,36,32,35,34,28);(5,6,11,18,9,13,8,10)(7,15,19,17,16,12,14,20)(22,27,30,29)(23,32)(28,35,36,31)(33,34);(2,4)(5,7,8,14,9,16,11,19)(6,12,10,17,13,15,18,20)(21,23,24,33,26,32,25,34)(22,28,29,35,30,36,27,31);(7,16)(12,15)(14,19)(17,20)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ch,d^c=dhi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72215867, #72224967, C2×#202645, … |
| maximal normal subgroup of | #72215867, #72224967, C2×#202645, … |
| maximal quotient of | #72224958, C2×#202645, #72322651, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109841);