Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309080 |
| 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 | #42887 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1607 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)(3,6)(4,5)(7,8)(17,18,23,27,22,28,21,37)(19,29,31,42,32,24,38,43)(20,33,35,45,36,26,39,25)(30,41,48,40,34,46,47,44);(2,5)(3,7)(10,14)(11,12)(17,19,22,32)(18,24,28,29)(20,34,36,30)(21,38,23,31)(25,44,45,40)(26,46,33,41)(27,43,37,42)(35,47,39,48);(9,10)(11,16)(12,15)(13,14)(17,20,22,36)(18,25,28,45)(19,30,32,34)(21,35,23,39)(24,40,29,44)(26,37,33,27)(31,47,38,48)(41,42,46,43);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,30)(21,39)(22,36)(23,35)(24,41)(25,27)(28,33)(29,46)(31,48)(32,34)(37,45)(38,47)(40,43)(42,44);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,27,28,37)(19,31,32,38)(20,35,36,39)(24,42,29,43)(25,26,45,33)(30,47,34,48)(40,46,44,41)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7397151);