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 | |
1368 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)(4,8)(10,14)(12,15)(18,22)(19,25)(24,29)(26,31)(33,34,39,42)(35,40,36,37)(38,41,46,48)(43,47,44,45);(2,3)(6,7)(10,14)(11,16)(19,25)(23,28)(26,31)(30,32)(33,35,39,36)(34,37,42,40)(38,44,46,43)(41,47,48,45);(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,36,39,35)(34,40,42,37)(38,44,46,43)(41,47,48,45);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,15,14,12)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,37,39,40)(34,36,42,35)(38,45,46,47)(41,44,48,43);(1,4,5,8)(2,6,3,7)(9,12,13,15)(10,16,14,11)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,38,39,46)(34,41,42,48)(35,43,36,44)(37,45,40,47)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=fh,e^2=g,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, 4394343);