Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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:
(11,15)(13,16)(17,18,21,22)(19,25,26,23)(20,24,28,30)(27,32,29,31)(33,34,37,40)(35,41,42,38)(36,44,45,39)(43,46,48,47);(1,2,5,8)(3,6,4,7)(18,22)(19,26)(20,28)(31,32)(35,42)(38,41)(43,48)(46,47);(1,2)(3,6)(4,7)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,30)(29,31)(33,34)(35,38)(36,39)(37,40)(41,42)(43,46)(44,45)(47,48);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,26)(18,23,22,25)(20,29,28,27)(24,32,30,31)(33,35)(34,38)(36,43)(37,42)(39,46)(40,41)(44,47)(45,48);(1,4,5,3)(2,7,8,6)(17,20,21,28)(18,24,22,30)(19,27,26,29)(23,31,25,32)(33,36,37,45)(34,39,40,44)(35,43,42,48)(38,46,41,47)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=gi,d^2=h,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3036827);