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 | |
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:
(3,7)(6,8)(9,10,14,12)(11,15,13,16)(17,18,21,24)(19,25,26,22)(20,23,28,30)(27,31,32,29)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,45)(46,48);(9,11,14,13)(10,15,12,16)(19,26)(22,25)(27,32)(29,31)(34,36)(37,45)(39,41)(42,48);(1,2,5,4)(3,6,7,8)(9,12,14,10)(11,15,13,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(1,3)(2,6)(4,8)(5,7)(17,19)(18,22)(20,27)(21,26)(23,29)(24,25)(28,32)(30,31)(33,36,38,34)(35,39,43,41)(37,44,45,40)(42,46,48,47);(1,4,5,2)(3,8,7,6)(9,13,14,11)(10,15,12,16)(17,20,21,28)(18,23,24,30)(19,27,26,32)(22,29,25,31)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fghi,d^2=f,e^2=fhi,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, 4496025);