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 | |
1370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,4)(6,8)(9,10,13,15)(11,12,16,14)(19,25)(20,27)(23,28)(24,29)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,47)(46,48);(1,2,5,4)(3,6,7,8)(9,11)(10,14)(12,15)(13,16)(18,22)(23,28)(24,29)(30,32)(34,35)(37,45)(39,41)(43,46);(3,7)(6,8)(9,10,13,15)(11,14,16,12)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,48,46,47);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36,38,44)(34,39,35,41)(37,46,45,43)(40,48,42,47);(1,4,5,2)(3,6,7,8)(9,12,13,14)(10,11,15,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,45)(34,40,35,42)(36,43,44,46)(39,47,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=fhi,c^2=gi,d^2=g,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, 5708117);