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 | #45136 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1488 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)(9,10)(11,15)(12,16)(13,14)(17,18)(19,28)(20,31)(21,26)(22,27)(23,24)(25,32)(29,41)(30,42)(33,47)(34,38)(35,39)(36,48)(37,44)(40,46)(43,45);(1,3)(2,6)(4,7)(5,8)(10,14)(11,12)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,35,37,30)(25,31,41,38)(26,42,44,39)(33,45,36,46)(40,48,43,47);(9,10)(11,16)(12,15)(13,14)(17,20,23,32,22,34,19,29)(18,25,28,31,27,41,24,38)(21,36,35,45,37,33,30,46)(26,43,42,48,44,40,39,47);(1,2,5,4)(3,7,8,6)(9,11,13,12)(10,15,14,16)(17,21,22,37)(18,26,27,44)(19,30,23,35)(20,33,34,36)(24,39,28,42)(25,40,41,43)(29,45,32,46)(31,47,38,48);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,15,14,16)(18,27)(20,34)(21,37)(24,28)(29,32)(30,35)(40,43)(47,48)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=g,d^2=f,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, 6957728);