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 | |
1364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,32,30,31)(33,34,38,41)(35,42,39,36)(37,45,46,40)(43,44,47,48);(9,10,13,12)(11,15,16,14)(18,19)(21,29)(23,25)(27,30)(35,39)(36,42)(43,47)(44,48);(1,2,5,4)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,31,30,32)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(1,3,5,7)(2,6,4,8)(17,20)(18,23)(19,25)(21,30)(22,28)(24,32)(26,31)(27,29)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,15,16,14)(17,21)(18,24)(19,26)(20,27)(22,29)(23,31)(25,32)(28,30)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=ghi,d^2=gi,e^2=fhi,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, 3230172);