Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1456 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)(3,8)(4,6)(5,7)(11,15)(14,16)(17,18,24,38,23,29,22,28)(19,30,37,25,34,44,33,41)(20,26,36,43,35,42,21,27)(31,45,32,39,46,40,47,48);(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,31)(21,32)(22,37)(23,34)(24,33)(26,39)(27,40)(28,44)(29,41)(30,38)(35,46)(36,47)(42,48)(43,45);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(17,20)(18,26)(19,31)(21,22)(23,35)(24,36)(25,39)(27,28)(29,42)(30,45)(32,37)(33,47)(34,46)(38,43)(40,44)(41,48);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,21,23,36)(18,27,29,43)(19,32,34,47)(20,22,35,24)(25,40,41,45)(26,28,42,38)(30,39,44,48)(31,37,46,33);(9,11,12,15)(10,14,13,16)(20,35)(21,36)(26,42)(27,43)(31,46)(32,47)(39,48)(40,45)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=gi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6403076);