Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(10,13)(11,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34)(35,39)(36,42)(37,40)(38,41)(43,46)(44,48)(45,47);(10,13)(14,16)(18,22)(23,25)(24,28)(30,31)(33,35,38,36)(34,39,41,42)(37,43,45,44)(40,46,47,48);(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32)(33,36,38,35)(34,39,41,42)(37,44,45,43)(40,46,47,48);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,37,38,45)(34,40,41,47)(35,43,36,44)(39,46,42,48)Polycyclic presentation:
2^9|a^2=hi,b^2=fgh,c^2=fgh,d^2=fhi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4768170);