Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,9,10,7)(4,8,12,13)(11,15,16,14)(19,26)(22,24)(27,32)(29,31)(33,34,40,38)(35,43,47,41)(36,37,42,39)(44,45,48,46);(2,6)(4,12)(7,9)(11,16)(17,18,23,20)(19,24,26,22)(21,28,30,25)(27,29,32,31)(33,35,40,47)(34,41,38,43)(36,48,42,44)(37,45,39,46);(3,10)(7,9)(11,16)(14,15)(17,19,23,26)(18,24,20,22)(21,27,30,32)(25,31,28,29)(33,36)(34,39)(35,44)(37,38)(40,42)(41,45)(43,46)(47,48);(1,3,5,10)(2,7,6,9)(4,11,12,16)(8,14,13,15)(17,20,23,18)(19,22,26,24)(21,28,30,25)(27,29,32,31)(33,37,40,39)(34,36,38,42)(35,45,47,46)(41,48,43,44);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21,23,30)(18,25,20,28)(19,27,26,32)(22,29,24,31)(33,38,40,34)(35,41,47,43)(36,37,42,39)(44,45,48,46);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,22)(18,19)(20,26)(21,29)(23,24)(25,27)(28,32)(30,31)(33,39,40,37)(34,42,38,36)(35,46,47,45)(41,44,43,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662598);