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) = C27 |
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 | |
1342 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)(5,8)(11,15)(13,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,34)(35,41)(36,38)(37,44)(39,43)(40,46)(42,47)(45,48);(1,2,4,6)(3,5,7,8)(18,19)(23,25)(24,26)(30,31)(34,36)(39,41)(40,44)(47,48);(1,2)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,19,22,18)(20,25,28,23)(21,24,29,26)(27,30,32,31)(33,35)(34,39)(36,41)(37,45)(38,43)(40,48)(42,46)(44,47);(9,11)(10,13)(12,15)(14,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,36,38,34)(35,41,43,39)(37,44,46,40)(42,48,45,47);(1,3)(2,5)(4,7)(6,8)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=gi,d^2=g,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, 4540018);