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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,3)(4,7,8,6)(9,10,12,14)(11,13,15,16)(18,22)(20,27)(23,28)(26,31)(33,34,37,38)(35,41,43,39)(36,44,46,40)(42,47,45,48);(9,11,12,15)(10,13,14,16)(18,22)(19,25)(24,29)(26,31)(34,38)(39,41)(40,44)(47,48);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,34)(35,39)(36,44)(37,38)(40,46)(41,43)(42,48)(45,47);(1,4,5,8)(2,6,3,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,46)(44,47);(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,36,37,46)(34,40,38,44)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=hi,d^2=fghi,e^2=f,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, 3461185);