Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885129 |
| 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×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,28)(20,32)(21,27)(22,26)(23,37)(24,30)(25,35)(29,44)(31,40)(33,42)(34,41)(36,39)(38,47)(43,45)(46,48);(1,2,4,5)(3,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,38)(26,28)(27,40)(32,48)(33,46)(34,47)(35,45)(37,39)(41,43)(42,44);(3,7)(6,8)(11,16)(13,15)(17,20)(18,25)(19,29)(21,35)(22,33)(23,34)(24,38)(26,41)(27,42)(28,43)(30,46)(31,47)(32,37)(36,45)(39,48)(40,44);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(18,26)(19,30)(20,33)(27,37)(31,36)(34,35)(38,43)(44,48);(2,5)(6,8)(17,21,22,23)(18,27,26,37)(19,31,30,36)(20,34,33,35)(24,39,28,40)(25,32,41,42)(29,45,46,47)(38,44,43,48)Polycyclic presentation:
2^9|b^2=fi,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei,g^c=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6838064);