Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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 | |
1542 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,27)(19,29,32,24)(20,25,23,28)(21,26,35,41)(30,43,45,37)(31,39,33,38)(34,40,36,42)(44,48,46,47);(2,6)(7,8)(17,19)(18,24)(20,33)(21,30)(22,32)(23,31)(25,39)(26,37)(27,29)(28,38)(34,46)(35,45)(36,44)(40,48)(41,43)(42,47);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20)(18,25)(21,35)(22,23)(26,41)(27,28)(30,44)(31,33)(37,47)(38,39)(43,48)(45,46);(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,30)(20,34)(22,35)(23,36)(24,37)(25,40)(27,41)(28,42)(29,43)(31,44)(32,45)(33,46)(38,47)(39,48);(1,4,5,3)(2,8,6,7)(19,31)(20,23)(21,36)(24,38)(25,28)(26,42)(29,39)(32,33)(34,35)(40,41)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=i,c^2=fi,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6494905);