Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993899 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,30)(20,35)(21,29)(22,28)(23,27)(24,33)(25,36)(26,37)(31,40)(32,43)(34,44)(38,41)(39,42)(45,46)(47,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,24)(20,31)(21,38)(22,33)(23,34)(25,40)(26,43)(27,41)(28,44)(29,30)(32,36)(35,45)(37,47)(39,46)(42,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,25)(19,31)(21,36)(22,39)(23,37)(24,40)(26,29)(27,42)(28,35)(30,43)(32,38)(33,46)(34,47)(41,48)(44,45);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20)(18,26)(19,32)(21,39)(22,36)(23,37)(24,40)(25,29)(27,35)(28,42)(30,45)(31,38)(33,47)(34,46)(41,48)(43,44);(2,8)(6,9)(7,16)(14,15)(18,27)(24,41)(25,42)(26,35)(28,29)(30,44)(40,48)(43,45)Polycyclic presentation:
2^9|c^2=h,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7031327);