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 | |
1378 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)(9,10,12,13)(11,15,16,14)(17,18,22,26)(19,20,24,23)(21,29,30,25)(27,31,32,28)(34,35)(39,41)(40,42)(46,47);(1,2)(3,5)(4,7)(6,8)(10,13)(11,16)(19,24)(20,23)(27,32)(28,31)(33,34,38,35)(36,41,44,39)(37,42,45,40)(43,46,48,47);(2,3)(6,7)(9,11,12,16)(10,14,13,15)(17,19,22,24)(18,23,26,20)(21,27,30,32)(25,31,29,28)(33,35,38,34)(36,39,44,41)(37,42,45,40)(43,46,48,47);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,32,29,27)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,21,22,30)(18,25,26,29)(19,27,24,32)(20,28,23,31)(33,37)(34,40)(35,42)(36,43)(38,45)(39,46)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=fhi,d^2=gi,e^2=fg,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, 4430768);