Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(11,16)(12,14)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)(33,34,38,40)(35,41,43,36)(37,39,45,46)(42,47,48,44);(1,2)(3,7)(4,6)(5,8)(9,10,13,15)(11,12,16,14)(18,22)(20,27)(23,29)(25,31)(37,45)(39,46)(42,48)(44,47);(1,2,4,6)(3,5,8,7)(9,11,13,16)(10,14,15,12)(17,18)(19,23)(20,28)(21,22)(24,27)(25,32)(26,29)(30,31)(33,35,38,43)(34,36,40,41)(37,42,45,48)(39,44,46,47);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19,21,26)(18,23,22,29)(20,25,27,31)(24,30,28,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,3)(2,5)(4,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,20)(18,24)(19,25)(21,27)(22,28)(23,30)(26,31)(29,32)(33,37,38,45)(34,39,40,46)(35,42,43,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=g,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5393144);