Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198798 |
| 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 | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6,10,14)(4,7)(5,8)(9,12)(11,13,16,15)(17,18,21,23,22,25,20,26)(19,27,29,24,30,32,28,31);(1,3)(2,6,8,14)(4,11)(5,10)(7,13,9,15)(12,16)(18,23)(20,21)(24,27)(25,26)(28,29)(31,32);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,19,22,30)(18,24,25,31)(20,29,21,28)(23,27,26,32);(2,8)(6,14)(7,9)(13,15)(18,25)(23,26)(24,31)(27,32)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=hi,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279569, #72281547, #72281573, … |
| maximal normal subgroup of | #72279569, #72281547, #72281573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106276);