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 | |
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:
(2,5)(6,8)(9,10,14,12)(11,15,13,16)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31)(33,34)(35,36)(37,42)(38,39)(40,41)(43,44)(45,48)(46,47);(9,11,14,13)(10,15,12,16)(19,26)(21,29)(23,25)(24,28)(34,39)(35,41)(42,47)(43,45);(9,12,14,10)(11,15,13,16)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31)(33,35)(34,40)(36,39)(37,45)(38,41)(42,44)(43,46)(47,48);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,16,12,15)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31)(33,36,38,40)(34,41,39,35)(37,44,46,48)(42,45,47,43);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,15,12,16)(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31)(33,37,38,46)(34,42,39,47)(35,43,41,45)(36,44,40,48)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=fh,d^2=h,e^2=fgi,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, 5463352);