Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(17,18,22,24)(19,28,29,41)(20,26,34,39)(21,27,23,36)(25,37,38,31)(30,45,32,48)(33,43,35,44)(40,46,42,47);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(18,24)(19,29)(26,39)(27,36)(30,32)(31,37)(43,44)(46,47);(10,13)(11,15)(17,19,22,29)(18,25,24,38)(20,32,34,30)(21,31,23,37)(26,42,39,40)(27,41,36,28)(33,47,35,46)(43,48,44,45);(9,11,12,15)(10,14,13,16)(17,20,22,34)(18,26,24,39)(19,30,29,32)(21,35,23,33)(25,40,38,42)(27,44,36,43)(28,45,41,48)(31,47,37,46);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,21,22,23)(18,27,24,36)(19,31,29,37)(20,33,34,35)(25,41,38,28)(26,43,39,44)(30,46,32,47)(40,45,42,48)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=h,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6422716);