Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| 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 | |
1300 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)(7,15,19,11)(8,20)(9,24)(10,13)(12,23)(14,25)(16,29,32,36)(17,27,18,28)(21,26,33,31)(22,34)(30,35);(1,3,4,2)(5,7)(6,11)(8,21)(9,17)(10,19)(12,29)(13,15)(14,27)(16,23)(18,25)(20,26)(22,33)(24,28)(30,36)(31,34)(32,35);(5,8,9,23,10,22,25,35)(6,12,14,20,13,30,24,34)(7,16,18,21,19,32,17,33)(11,26,28,29,15,31,27,36);(5,7)(6,11)(8,16)(9,18)(10,19)(12,26)(13,15)(14,28)(17,25)(20,29)(21,23)(22,32)(24,27)(30,31)(33,35)(34,36)Polycyclic presentation:
2^9|a^2=h,b^2=fghi,c^2=g,f^2=i,g^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^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280919, #72280941, #72281258, … |
| maximal normal subgroup of | #72280919, #72280941, #72281258, … |
| maximal quotient of | #72313358 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277310);