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 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,8,4)(9,10,13,15)(11,16,14,12)(18,22)(19,25)(24,29)(27,31)(33,34)(35,42)(36,40)(37,41)(38,39)(43,48)(44,45)(46,47);(1,3)(2,4)(5,8)(6,7)(9,11,13,14)(10,12,15,16)(20,28)(24,29)(27,31)(30,32)(34,39)(35,40)(41,47)(43,45);(2,6)(4,7)(9,10)(11,12)(13,15)(14,16)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35)(34,36)(37,45)(38,40)(39,42)(41,48)(43,46)(44,47);(1,4)(2,3)(5,7)(6,8)(9,12)(10,14)(11,15)(13,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36)(34,40)(35,39)(37,44)(38,42)(41,45)(43,47)(46,48);(1,2)(3,4)(5,6)(7,8)(9,12)(10,14)(11,15)(13,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37,38,46)(34,41,39,47)(35,43,40,45)(36,44,42,48)Polycyclic presentation:
2^9|a^2=fh,b^2=fgi,c^2=fg,d^2=fg,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, 5393438);