Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,15,18,11)(8,19)(9,13)(10,14)(12,21)(16,31,32,25)(17,30,24,26)(20,33,35,27)(22,34)(23,29)(28,36);(1,3,4,2)(5,7)(6,11)(8,20)(9,18)(10,24)(12,27)(13,15)(14,30)(16,22)(17,23)(19,33)(21,35)(25,28)(26,29)(31,34)(32,36);(5,8,10,22,9,21,23,36)(6,12,14,28,13,19,29,34)(7,16,17,20,18,32,24,35)(11,25,26,27,15,31,30,33);(5,7)(6,11)(8,16)(9,18)(10,17)(12,25)(13,15)(14,26)(19,31)(20,22)(21,32)(23,24)(27,28)(29,30)(33,34)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=g,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,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 | #72280941, #72281191, #72281232, … |
| maximal normal subgroup of | #72280941, #72281191, #72281232, … |
| maximal quotient of | #72313400 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275662);