Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1410 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)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(20,27)(24,29)(25,31)(30,32)(33,34)(35,39)(36,43)(37,38)(40,44)(41,47)(42,45)(46,48);(4,8)(6,7)(9,11,13,16)(10,14,12,15)(18,22)(23,28)(24,29)(30,32)(34,37)(39,45)(40,43)(46,47);(2,3)(6,7)(10,12)(14,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,35,38,42)(34,39,37,45)(36,41,44,48)(40,46,43,47);(1,3,5,2)(4,7,8,6)(9,12,13,10)(11,15,16,14)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36)(34,40)(35,41)(37,43)(38,44)(39,46)(42,48)(45,47);(1,4)(2,6)(3,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37,38,34)(35,39,42,45)(36,43,44,40)(41,46,48,47)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=f,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5394718);