Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,4)(6,7)(9,10,13,11)(12,15,16,14)(17,18)(19,22)(20,23)(21,26)(24,29)(25,27)(28,31)(30,32)(33,34,37,38)(35,39,41,42)(36,44,46,40)(43,48,45,47);(2,4)(6,7)(9,11,13,10)(12,15,16,14)(18,19)(20,27)(21,29)(30,31)(34,38)(35,41)(40,44)(43,45);(1,2,5,4)(3,7,8,6)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,35)(34,39)(36,45)(37,41)(38,42)(40,48)(43,46)(44,47);(1,3,5,8)(2,6,4,7)(9,12)(10,14)(11,15)(13,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,34)(35,42)(36,40)(37,38)(39,41)(43,48)(44,46)(45,47);(1,4,5,2)(3,7,8,6)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,36)(34,40)(35,43)(37,46)(38,44)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=g,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4976323);