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 | |
1352 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)(11,12)(15,16)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,31,32,30)(33,34,38,36)(35,41,43,39)(37,44,46,40)(42,47,45,48);(2,5)(6,8)(10,14)(15,16)(18,19)(21,29)(23,25)(27,32)(33,35,38,43)(34,39,36,41)(37,45,46,42)(40,48,44,47);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(9,11,13,12)(10,15,14,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,36,38,34)(35,39,43,41)(37,44,46,40)(42,47,45,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ghi,d^2=gi,e^2=gh,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, 4433517);