Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(10,12)(11,15)(18,22)(19,25)(24,29)(27,31)(34,36)(35,41)(37,45)(47,48);(10,12)(14,16)(20,28)(24,29)(27,31)(30,32)(33,34,38,36)(35,39,41,43)(37,40,45,44)(42,47,46,48);(9,10,13,12)(11,16,15,14)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35)(34,39)(36,43)(37,46)(38,41)(40,48)(42,45)(44,47);(1,2)(3,5)(4,6)(7,8)(9,11,13,15)(10,14,12,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,38,34)(35,39,41,43)(37,44,45,40)(42,47,46,48);(1,3,4,7)(2,5,6,8)(9,12,13,10)(11,16,15,14)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,47)(41,46)(43,48)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=h,d^2=fh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5509389);