Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1493 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,20)(19,26,30,27)(21,25,22,32)(24,28,31,29);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,31)(21,29)(22,28)(23,30)(25,27)(26,32);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,13,14,15)(17,18)(19,24)(20,23)(21,32)(22,25)(26,29)(27,28)(30,31);(11,16)(13,15)(17,20,23,18)(19,27,30,26)(21,32,22,25)(24,29,31,28);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(17,21,23,22)(18,25,20,32)(19,28,30,29)(24,26,31,27)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=ghi,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7000409);