Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #368859 |
| 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 | C2×#3068 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1689 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)(4,17,21,31)(5,14)(6,23)(7,10)(8,11)(9,20,26,19)(13,25,30,18)(15,22)(16,24)(27,29,32,28);(1,3)(2,5)(4,13,21,30)(6,15)(7,14)(8,16)(9,25,26,18)(10,24)(11,22)(12,23)(17,27,31,32)(19,29,20,28);(1,4,6,19,8,21,7,20)(2,9,10,17,11,26,23,31)(3,13,15,29,16,30,14,28)(5,18,24,32,22,25,12,27);(1,5)(2,3)(4,18)(6,24)(7,12)(8,22)(9,13)(10,15)(11,16)(14,23)(17,29)(19,32)(20,27)(21,25)(26,30)(28,31)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,e^2=i,f^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,g^b=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72316368, C2×#368859, #72319403 |
| maximal normal subgroup of | #72316368, C2×#368859, #72319403 |
| maximal quotient of | C2×#368859, #72316831 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276055);