Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| 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 | #42908 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1779 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)(3,7)(4,6)(5,8)(17,18,24,29)(19,25,32,43)(20,27,21,35)(22,28,23,38)(26,36,45,34)(30,46,33,40)(31,41,37,42)(39,47,44,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,33)(21,30)(22,37)(23,31)(24,32)(26,44)(27,40)(28,42)(29,43)(34,47)(35,46)(36,48)(38,41)(39,45);(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,30)(21,24)(22,34)(23,36)(25,39)(27,38)(28,35)(29,45)(31,47)(32,33)(37,48)(40,42)(41,46)(43,44);(3,8)(5,7)(11,16)(13,15)(17,21,24,20)(18,27,29,35)(19,30,32,33)(22,34,23,36)(25,40,43,46)(26,28,45,38)(31,48,37,47)(39,41,44,42);(2,6)(5,7)(10,14)(13,15)(17,22,24,23)(18,28,29,38)(19,31,32,37)(20,34,21,36)(25,41,43,42)(26,35,45,27)(30,47,33,48)(39,46,44,40)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7141700);