Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #23853 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,27,29,20)(21,24,30,32)(23,31,28,26);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,22,29)(18,20,25,27)(21,23,30,28)(24,26,32,31);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20,28,24,22,27,23,32)(18,21,31,29,25,30,26,19);(2,6)(7,8)(11,16)(13,15)(17,21,22,30)(18,24,25,32)(19,28,29,23)(20,31,27,26);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(18,25)(19,29)(21,30)(26,31)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=i,e^2=fi,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^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6934534);