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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4256 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,4,10,8)(5,13,14,9)(11,15,16,12)(20,28)(24,29)(26,32)(30,31)(33,34,38,42)(35,39,36,40)(37,45,46,41)(43,48,44,47);(2,7)(3,10)(9,13)(11,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(2,7)(3,10)(9,13)(11,16)(18,22)(23,25)(24,29)(30,31)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,48)(44,47)(45,46);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36,38,35)(34,40,42,39)(37,44,46,43)(41,48,45,47);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,46)(34,41,42,45)(35,43,36,44)(39,47,40,48)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8062246);