Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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 | |
1364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,3)(6,7)(10,14)(15,16)(17,18,21,24)(19,25,26,22)(20,28,29,23)(27,30,32,31)(34,35)(36,43)(40,42)(44,48);(19,26)(22,25)(27,32)(30,31)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,47,48,46);(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32)(33,35,38,34)(36,39,43,41)(37,42,45,40)(44,46,48,47);(1,3,5,2)(4,7,8,6)(9,11,13,12)(10,15,14,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,4,5,8)(2,6,3,7)(9,12,13,11)(10,16,14,15)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=hi,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4238032);