Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1306 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)(3,6)(4,5)(7,8)(11,15)(13,16)(17,18,21,22)(19,25,27,23)(20,28,30,24)(26,31,29,32)(34,38)(35,41)(40,44)(43,47);(2,5)(6,7)(18,22)(23,25)(24,28)(31,32)(33,34,37,38)(35,42,41,39)(36,40,45,44)(43,48,47,46);(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,13,15,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,48)(46,47);(9,11)(10,13)(12,15)(14,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,35,37,41)(34,39,38,42)(36,43,45,47)(40,46,44,48);(1,3,4,8)(2,6,5,7)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=gi,d^2=fgh,e^2=fg,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, 4544772);