Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14823 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,7,18,20)(3,11,12,21,14,8,19,23)(4,15,17,29,16,9,24,25)(13,26,27,31,28,22,30,32);(2,7)(3,12)(5,18)(8,21)(9,15)(11,23)(13,27)(14,19)(17,24)(22,31)(26,32)(28,30);(1,3)(2,8)(4,13)(5,19)(6,14)(7,11)(9,22)(10,21)(12,18)(15,26)(16,28)(17,30)(20,23)(24,27)(25,31)(29,32);(1,4)(2,9,7,15)(3,13)(5,17)(6,16)(8,22,11,26)(10,25,20,29)(12,27)(14,28)(18,24)(19,30)(21,31,23,32)Polycyclic presentation:
2^9|a^2=g,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308996);