Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #368855 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14023 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60426 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,17,21,26)(5,14)(6,10)(7,11)(8,20,27,32)(9,24)(13,19,30,31)(15,23)(16,22)(18,28,29,25);(1,3)(2,5)(4,18,21,29)(6,15)(7,16)(8,25,27,28)(9,22)(10,12)(11,23)(13,20,30,32)(14,24)(17,19,26,31);(1,4,7,21)(2,8)(3,13)(5,19,23,31)(6,20,24,32)(9,26)(10,17)(11,27)(12,28,22,25)(14,29)(15,18)(16,30);(1,5)(2,3)(4,19)(6,22)(7,23)(8,13)(9,15)(10,14)(11,16)(12,24)(17,29)(18,26)(20,25)(21,31)(27,30)(28,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72213409, #72216029, #72216123, … |
| maximal normal subgroup of | #72213409, #72216029, #72216123, … |
| maximal quotient of | C2×#368855, #72239641, #72239644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276051);