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 | |
1354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10)(11,16)(12,15)(13,14)(18,22)(19,25)(20,27)(30,32)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(1,2)(3,8)(4,5)(6,7)(9,11,14,12)(10,15,13,16)(18,22)(19,25)(24,29)(26,31)(36,45)(40,44)(42,48)(46,47);(2,5)(6,8)(11,12)(15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,43)(34,39,37,41)(36,42,45,48)(40,46,44,47);(1,2)(3,6)(4,5)(7,8)(9,12,14,11)(10,15,13,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,15,12,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,34)(35,41,43,39)(36,44,45,40)(42,47,48,46)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgi,c^2=g,d^2=fgi,e^2=gh,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, 4419639);