Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1625 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)(4,6)(5,8)(9,10,12,13)(11,14,15,16)(17,18,22,27)(19,28,32,24)(20,33,36,25)(21,26,38,23)(29,41,47,45)(30,39,42,31)(34,43,46,48)(35,40,44,37);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19,22,32)(18,24,27,28)(20,34,36,46)(21,30,38,42)(23,39,26,31)(25,43,33,48)(29,40,47,37)(35,45,44,41);(10,13)(14,16)(17,20,22,36)(18,25,27,33)(19,29,32,47)(21,35,38,44)(23,40,26,37)(24,41,28,45)(30,43,42,48)(31,46,39,34);(2,6)(5,7)(10,13)(14,16)(17,21)(18,26)(19,30)(20,35)(22,38)(23,27)(24,31)(25,37)(28,39)(29,43)(32,42)(33,40)(34,45)(36,44)(41,46)(47,48);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(19,31)(21,38)(23,26)(24,42)(28,30)(29,46)(32,39)(34,47)(35,44)(37,40)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6625645);