Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,4,7)(9,10,14,12)(11,15,13,16)(18,22)(19,25)(20,28)(30,32)(33,34,38,39)(35,40,41,36)(37,45,47,42)(43,44,46,48);(9,11,14,13)(10,15,12,16)(18,22)(20,28)(23,26)(27,31)(34,39)(36,40)(42,45)(44,48);(1,3,5,4)(2,6,8,7)(9,12,14,10)(11,15,13,16)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35)(34,40)(36,39)(37,46)(38,41)(42,44)(43,47)(45,48);(1,2)(3,6)(4,7)(5,8)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,38,40)(34,41,39,35)(37,44,47,48)(42,46,45,43);(1,4,5,3)(2,7,8,6)(9,13,14,11)(10,15,12,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,47)(34,42,39,45)(35,43,41,46)(36,44,40,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=fghi,d^2=fg,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5372199);