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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1735 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,33,44)(20,34,36,25)(21,26,37,43)(22,38,35,27)(24,39,42,32)(30,41,46,45)(31,40,47,48);(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,30)(21,31)(22,39)(23,33)(25,40)(26,41)(27,44)(28,42)(29,38)(32,35)(34,48)(36,46)(37,47)(43,45);(2,5)(6,8)(11,15)(13,16)(17,20,23,36)(18,25,28,34)(19,30,33,46)(21,35,37,22)(24,40,42,48)(26,38,43,27)(29,45,44,41)(31,32,47,39);(11,15)(13,16)(17,21)(18,26)(19,31)(20,35)(22,36)(23,37)(24,41)(25,38)(27,34)(28,43)(29,40)(30,32)(33,47)(39,46)(42,45)(44,48);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(20,36)(21,37)(25,34)(26,43)(30,46)(31,47)(40,48)(41,45)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,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, 6383496);