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 | |
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:
(1,2,5,8)(3,6,7,4)(9,10,13,11)(12,15,16,14)(18,22)(20,28)(23,25)(26,32)(33,34,38,41)(35,39,36,43)(37,40,45,48)(42,46,44,47);(1,2)(3,6)(4,7)(5,8)(18,22)(20,28)(23,25)(26,32)(33,35,38,36)(34,39,41,43)(37,44,45,42)(40,47,48,46);(1,3,5,7)(2,6,8,4)(9,11,13,10)(12,15,16,14)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,36,38,35)(34,39,41,43)(37,44,45,42)(40,46,48,47);(1,3)(2,6)(4,8)(5,7)(9,12,13,16)(10,14,11,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,34)(35,39)(36,43)(37,40)(38,41)(42,46)(44,47)(45,48);(1,4,5,6)(2,7,8,3)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,45)(34,40,41,48)(35,42,36,44)(39,46,43,47)Polycyclic presentation:
2^9|a^2=fi,b^2=gh,c^2=fhi,d^2=gi,e^2=fgh,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, 5787576);