Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,3)(4,8)(11,15)(14,16)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32)(33,34,38,41)(35,42,39,36)(37,45,46,40)(43,44,47,48);(1,2,5,3)(4,7,8,6)(18,19)(20,27)(21,29)(30,31)(35,39)(36,42)(43,47)(44,48);(2,3)(6,7)(9,10,13,12)(11,14,15,16)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,16,15,14)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=ghi,d^2=gh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4504477);