Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,3)(4,6,8,7)(11,15)(13,16)(19,25)(20,27)(23,28)(24,29)(33,34)(35,40)(36,42)(37,41)(38,39)(43,45)(44,48)(46,47);(4,8)(6,7)(9,10,12,14)(11,13,15,16)(18,22)(23,28)(24,29)(30,32)(34,39)(35,42)(41,47)(43,48);(2,3)(6,7)(9,11,12,15)(10,13,14,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35,38,42)(34,40,39,36)(37,43,46,48)(41,45,47,44);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36,38,40)(34,35,39,42)(37,45,46,44)(41,48,47,43);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,45)(42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=fhi,d^2=fghi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5861791);