Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ 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:
(1,2,5,6)(3,7,4,8)(17,18,22,27)(19,28,30,40)(20,32,36,42)(21,25,33,26)(23,37,39,31)(24,34,41,35)(29,44,45,38)(43,46,48,47);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,31,22,30,33,37)(18,23,26,28,27,39,25,40)(20,29,35,47,36,45,34,46)(24,38,32,43,41,44,42,48);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,33,34,22,36,21,35)(18,24,25,42,27,41,26,32)(19,29,37,46,30,45,31,47)(23,38,40,48,39,44,28,43);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(19,30)(23,39)(28,40)(29,45)(31,37)(38,44)(43,48)(46,47);(11,15)(14,16)(17,21,22,33)(18,25,27,26)(19,31,30,37)(20,34,36,35)(23,40,39,28)(24,32,41,42)(29,46,45,47)(38,43,44,48)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=fi,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6464666);