Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(10,13)(11,15)(17,18,22,26)(19,27,28,40)(20,24,35,44)(21,36,37,25)(23,30,38,31)(29,45,32,48)(33,43,34,42)(39,47,41,46);(1,3,5,8)(2,6,4,7)(9,10,12,13)(11,16,15,14)(19,28)(23,38)(27,40)(29,32)(30,31)(39,41)(45,48)(46,47);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,31,22,28,37,30)(18,23,25,27,26,38,36,40)(20,32,34,47,35,29,33,46)(24,41,43,48,44,39,42,45);(3,8)(6,7)(10,13)(14,16)(17,20,22,35)(18,24,26,44)(19,29,28,32)(21,34,37,33)(23,39,38,41)(25,43,36,42)(27,45,40,48)(30,47,31,46);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(17,21,22,37)(18,25,26,36)(19,30,28,31)(20,33,35,34)(23,40,38,27)(24,42,44,43)(29,46,32,47)(39,45,41,48)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=fg,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6956698);