Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(18,22)(19,25)(20,27)(30,32)(33,34,38,39)(35,40,41,36)(37,42,45,46)(43,47,48,44);(1,2,5,4)(3,6,7,8)(9,10,13,16)(11,14,12,15)(18,22)(19,25)(24,29)(26,31)(34,39)(35,41)(37,45)(44,47);(1,2,5,4)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,35)(34,40)(36,39)(37,43)(38,41)(42,47)(44,46)(45,48);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,14,16,15)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36,38,40)(34,41,39,35)(37,44,45,47)(42,48,46,43);(1,4,5,2)(3,8,7,6)(9,12,13,11)(10,15,16,14)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48)Polycyclic presentation:
2^9|a^2=f,b^2=hi,c^2=fgi,d^2=h,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1478939);