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 | |
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:
(1,2,5,8)(3,6,4,7)(17,18,22,23)(19,26,24,20)(21,29,32,25)(27,28,30,31)(33,34,37,41)(35,39,43,38)(36,40,46,45)(42,48,44,47);(10,13)(14,16)(18,23)(19,24)(25,29)(27,30)(33,35,37,43)(34,38,41,39)(36,42,46,44)(40,47,45,48);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(17,19,22,24)(18,20,23,26)(21,30,32,27)(25,31,29,28)(33,35)(34,39)(36,44)(37,43)(38,41)(40,47)(42,46)(45,48);(1,2)(3,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,23)(21,31)(22,26)(25,27)(28,32)(29,30)(33,34)(35,39)(36,45)(37,41)(38,43)(40,46)(42,47)(44,48);(1,4,5,3)(2,7,8,6)(17,21)(18,25)(19,27)(20,28)(22,32)(23,29)(24,30)(26,31)(33,36)(34,40)(35,42)(37,46)(38,47)(39,48)(41,45)(43,44)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=ghi,d^2=fh,e^2=fi,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, 3613891);