Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23850 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,25)(22,26)(23,24)(27,28)(29,30)(31,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,26,20)(21,29,31,27)(25,28,32,30);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20,19,18,22,24,23,26)(21,30,27,32,31,28,29,25);(2,6)(7,8)(11,16)(13,15)(17,21)(18,25)(19,27)(20,28)(22,31)(23,29)(24,30)(26,32);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,19,22,23)(18,20,26,24)(21,29,31,27)(25,30,32,28)Polycyclic presentation:
2^9|b^2=i,c^2=fg,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6934579);