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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(17,18,22,27)(19,23,30,38)(20,25,32,41)(21,24,34,26)(28,36,31,40)(29,35,44,37)(33,39,46,42)(43,47,45,48);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,27,38)(20,31,32,28)(21,29,34,44)(24,35,26,37)(25,40,41,36)(33,45,46,43)(39,48,42,47);(10,13)(11,15)(18,24)(21,34)(23,35)(25,39)(26,27)(29,44)(33,46)(36,47)(37,38)(40,48)(41,42)(43,45);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,28)(21,33)(22,32)(23,36)(24,39)(26,42)(27,41)(29,43)(30,31)(34,46)(35,47)(37,48)(38,40)(44,45);(1,4,5,3)(2,6,8,7)(10,13)(14,16)(20,32)(25,41)(28,31)(33,46)(36,40)(39,42)(43,45)(47,48)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,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, 6423864);