Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #197339 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#2631 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,6,13,4,21)(3,15,18,31,19,26,32,30)(5,17,16,20,25,11,22,29)(8,14,27,24,9,28,23,10);(1,3,16,13,8,20,30,28)(2,9,29,31,14,6,19,22)(4,17,5,24,23,32,26,12)(7,11,25,10,27,18,15,21);(4,7)(5,25)(10,24)(11,17)(12,21)(15,26)(18,32)(23,27);(1,4,6,7)(2,10,13,24)(3,17,19,11)(5,22,25,16)(8,23,9,27)(12,14,21,28)(15,30,26,31)(18,20,32,29)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=eh,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213796, #72213906, #72219481, … |
| maximal normal subgroup of | #72213796, #72213906, #72219481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 104584);