Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1633 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)(3,6)(4,7)(5,8)(17,18)(19,23)(20,25)(21,24)(22,27)(26,34)(28,36)(29,35)(30,38)(31,40)(32,41)(33,39)(37,44)(42,46)(43,47)(45,48);(1,3,6,2)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,27,38)(20,31,32,28)(21,29,34,44)(24,35,26,37)(25,40,41,36)(33,45,46,43)(39,48,42,47);(9,11)(10,13)(12,16)(14,15)(18,24)(21,34)(23,35)(25,39)(26,27)(29,44)(33,46)(36,47)(37,38)(40,48)(41,42)(43,45);(1,4,6,5)(2,7,3,8)(10,14)(11,16)(17,20)(18,25)(19,28)(21,33)(22,32)(23,36)(24,39)(26,42)(27,41)(29,43)(30,31)(34,46)(35,47)(37,48)(38,40)(44,45);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(20,32)(25,41)(28,31)(33,46)(36,40)(39,42)(43,45)(47,48)Polycyclic presentation:
2^9|b^2=fi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7215448);