Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1581 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(11,15)(14,16)(17,18,22,25)(19,27,31,40)(20,24,32,38)(21,34,35,26)(23,29,37,30)(28,41,43,48)(33,46,45,39)(36,42,47,44);(10,13)(11,15)(17,19,22,31)(18,23,25,37)(20,28,32,43)(21,29,35,30)(24,36,38,47)(26,40,34,27)(33,42,45,44)(39,48,46,41);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,32)(18,24,25,38)(19,28,31,43)(21,33,35,45)(23,36,37,47)(26,39,34,46)(27,41,40,48)(29,42,30,44);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,25)(19,29)(21,35)(23,27)(24,38)(28,42)(30,31)(33,45)(36,41)(37,40)(43,44)(47,48);(9,11,12,15)(10,14,13,16)(20,32)(24,38)(28,43)(33,45)(36,47)(39,46)(41,48)(42,44)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=i,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7068159);