Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,10,15,11,12,9,22)(7,13,16,26,18,25,23,28)(8,19,21,29,20,14,24,30)(17,27,33,36,32,34,31,35);(1,3,4,2)(6,12)(7,16)(9,10)(13,26)(14,19)(17,31)(18,23)(21,24)(25,28)(27,35)(32,33)(34,36);(5,7,9,16,11,18,10,23)(6,13,15,28,12,25,22,26)(8,17,21,31,20,32,24,33)(14,27,29,36,19,34,30,35);(5,8)(6,14)(7,17)(9,21)(10,24)(11,20)(12,19)(13,27)(15,29)(16,31)(18,32)(22,30)(23,33)(25,34)(26,35)(28,36)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=g,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72227326, #72312162, #72312223, … |
| maximal normal subgroup of | #72227326, #72312162, #72312223, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277313);