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 | |
1312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10,13,12)(11,14,15,16)(17,18)(19,20)(21,29)(22,23)(24,30)(25,28)(26,32)(27,31)(34,36)(35,41)(37,45)(46,48);(2,5)(6,8)(10,12)(14,16)(18,23)(19,25)(21,30)(27,32)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(11,15)(14,16)(17,19,22,25)(18,20,23,28)(21,26,30,31)(24,27,29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,31)(30,32)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,16,15,14)(17,21)(18,24)(19,26)(20,27)(22,30)(23,29)(25,31)(28,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=h,d^2=g,e^2=fgh,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, 5885199);