Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #368888 |
| 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 | |
1708 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)(3,15)(4,18,23,10)(5,24)(6,14)(7,12)(8,13)(9,17)(11,25)(16,27,28,31)(19,26,32,20)(21,29,22,30);(1,3)(2,9)(4,19,23,32)(5,13)(6,24)(7,17)(8,11)(10,27,18,31)(12,25)(14,15)(16,22,28,21)(20,29,26,30);(1,4,7,22,8,23,6,21)(2,10,12,29,13,18,14,30)(3,16,24,32,11,28,17,19)(5,20,25,31,9,26,15,27);(1,5)(2,11)(3,13)(4,20)(6,15)(7,25)(8,9)(10,28)(12,17)(14,24)(16,18)(19,29)(21,27)(22,31)(23,26)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,e^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,e^b=ei,f^b=fi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215995, #72216130, #72216217, … |
| maximal normal subgroup of | #72215995, #72216130, #72216217, … |
| maximal quotient of | C2×#368888, #72252140, #72252177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276084);