Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,16)(12,13)(14,15)(20,27)(24,29)(25,31)(30,32)(33,34)(35,39)(36,38)(37,44)(40,43)(41,45)(42,47)(46,48);(10,15)(11,13)(18,22)(23,28)(24,29)(30,32)(33,35,38,43)(34,39,36,40)(37,42,45,48)(41,46,44,47);(9,11,14,13)(10,16,15,12)(17,18,21,22)(19,23,26,28)(20,24,27,29)(25,30,31,32)(33,36,38,34)(35,39,43,40)(37,44,45,41)(42,46,48,47);(1,2)(3,5)(4,6)(7,8)(9,12,14,16)(10,11,15,13)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,3,4,7)(2,5,6,8)(9,13,14,11)(10,16,15,12)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37)(34,41)(35,42)(36,44)(38,45)(39,46)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=h,d^2=gh,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4462365);