Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200140 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60799 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1321 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,13,18,28,3,15)(4,20,23,30,32,12,19,27)(6,14,16,8,7,10,17,11)(9,26,24,22,29,31,25,21);(2,8,14,13)(3,16)(4,21)(5,17)(9,24,20,30)(10,15,28,11)(12,27,29,25)(19,23)(22,32)(26,31);(2,9)(3,17)(4,22)(5,16)(8,24)(10,12)(11,25)(13,30)(14,20)(15,27)(19,23)(21,32)(26,31)(28,29);(2,10)(8,15)(9,12)(11,13)(14,28)(20,29)(24,27)(25,30)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213382, #72213874, #72219481, … |
| maximal normal subgroup of | #72213382, #72213874, #72219481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107548);