Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1435 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(17,18,21,22)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,46)(45,47);(10,13)(14,16)(18,22)(23,25)(24,28)(31,32)(33,34,37,40)(35,41,43,38)(36,39,46,44)(42,48,45,47);(1,2,5,4)(3,6,7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(1,3,5,7)(2,6,4,8)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,35)(34,38)(36,45)(37,43)(39,48)(40,41)(42,46)(44,47);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)(33,36)(34,39)(35,42)(37,46)(38,47)(40,44)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=hi,d^2=fi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3720787);