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 | |
1300 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)(9,10,13,11)(12,15,16,14)(18,22)(19,25)(24,29)(27,31)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(2,5)(6,8)(9,11,13,10)(12,15,16,14)(20,28)(24,29)(27,31)(30,32)(34,35)(39,41)(40,42)(46,47);(1,2)(3,6)(4,5)(7,8)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,38,34)(36,39,44,41)(37,40,45,42)(43,47,48,46);(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,3,4,7)(2,6,5,8)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=f,d^2=f,e^2=h,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, 5395791);