Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10,14,15)(11,16,13,12)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32)(33,34,38,40)(35,36,42,44)(37,39,45,46)(41,43,47,48);(2,4)(3,8)(10,15)(11,13)(18,22)(19,26)(20,28)(30,31)(33,35,38,42)(34,36,40,44)(37,41,45,47)(39,43,46,48);(1,3)(2,6)(4,7)(5,8)(9,11,14,13)(10,16,15,12)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32)(33,35)(34,36)(37,41)(38,42)(39,43)(40,44)(45,47)(46,48);(1,4,5,2)(3,7,8,6)(9,12,14,16)(10,11,15,13)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,3)(2,6)(4,7)(5,8)(9,13,14,11)(10,16,15,12)(17,20,21,28)(18,24,22,29)(19,27,26,32)(23,30,25,31)(33,37,38,45)(34,39,40,46)(35,41,42,47)(36,43,44,48)Polycyclic presentation:
2^9|a^2=fi,b^2=fh,c^2=ghi,d^2=hi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6038130);