Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1943 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,25)(21,34)(22,27)(23,30)(24,32)(26,36)(29,44)(31,42)(33,40)(35,39)(37,48)(38,46)(41,47)(43,45);(9,11)(10,13)(12,15)(14,16)(17,19,22,31)(18,23,27,39)(20,29,32,45)(21,35,36,30)(24,37,25,38)(26,42,34,28)(33,48,47,46)(40,44,41,43);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,20)(18,24)(19,29)(21,33)(22,32)(23,37)(25,27)(26,40)(28,43)(30,46)(31,45)(34,41)(35,48)(36,47)(38,39)(42,44);(11,15)(13,16)(17,20)(18,25)(19,29)(21,33)(22,32)(23,38)(24,27)(26,41)(28,44)(30,46)(31,45)(34,40)(35,48)(36,47)(37,39)(42,43);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(20,32)(24,25)(29,45)(33,47)(37,38)(40,41)(43,44)(46,48)Polycyclic presentation:
2^9|b^2=i,c^2=gi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6383966);