Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324500 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1741 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,23)(20,31)(21,25)(22,27)(24,28)(26,30)(29,39)(32,41)(33,47)(34,38)(35,42)(36,48)(37,44)(40,45)(43,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,27,26)(20,32,34,28)(21,35,37,29)(24,31,41,38)(25,42,44,39)(33,45,36,46)(40,48,43,47);(2,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,30,32,22,34,19,28)(18,24,23,38,27,41,26,31)(21,36,35,45,37,33,29,46)(25,43,39,47,44,40,42,48);(3,8)(5,7)(10,14)(11,16)(17,21)(18,25)(19,29)(20,33)(22,37)(23,39)(24,40)(26,42)(27,44)(28,45)(30,35)(31,47)(32,46)(34,36)(38,48)(41,43);(2,6)(5,7)(10,14)(13,15)(20,34)(21,37)(24,41)(25,44)(28,32)(29,35)(31,38)(39,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7234172);