Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1568 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)(17,18,22,27)(19,29,32,24)(20,33,35,42)(21,28,23,26)(25,37,40,34)(30,44,46,48)(31,39,36,41)(38,47,43,45);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,13)(14,16)(17,19,22,32)(18,24,27,29)(20,30,35,46)(21,36,23,31)(25,38,40,43)(26,41,28,39)(33,48,42,44)(34,47,37,45);(3,8)(5,7)(10,13)(14,15)(17,20)(18,25)(19,30)(21,34)(22,35)(23,37)(24,38)(26,33)(27,40)(28,42)(29,43)(31,45)(32,46)(36,47)(39,44)(41,48);(2,6)(5,7)(17,21,22,23)(18,26,27,28)(19,31,32,36)(20,34,35,37)(24,39,29,41)(25,33,40,42)(30,45,46,47)(38,44,43,48);(9,11,12,16)(10,14,13,15)(18,27)(19,32)(25,40)(26,28)(30,46)(31,36)(33,42)(45,47)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6699768);