Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
1364 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)(3,8)(4,6)(5,7)(9,10)(11,12)(13,15)(14,16)(17,18,21,24)(19,25,26,22)(20,28,29,23)(27,30,32,31)(33,34)(35,36)(37,40)(38,39)(41,44)(42,43)(45,46)(47,48);(9,11,14,13)(10,12,16,15)(19,26)(22,25)(27,32)(30,31)(34,39)(35,41)(37,45)(43,48);(1,3,6,5)(2,7,4,8)(9,11,14,13)(10,15,16,12)(17,19)(18,22)(20,27)(21,26)(23,30)(24,25)(28,31)(29,32)(33,35,38,41)(34,36,39,44)(37,42,45,47)(40,43,46,48);(1,4,6,2)(3,7,5,8)(9,12,14,15)(10,11,16,13)(17,18)(19,25)(20,23)(21,24)(22,26)(27,31)(28,29)(30,32)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,5,6,3)(2,7,4,8)(9,13,14,11)(10,15,16,12)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,43,44,48)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=gh,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4861855);