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 | |
4072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,5)(3,6,9,11)(4,8,13,14)(10,12,15,16)(17,18,24,27)(19,28,30,25)(20,22,21,31)(23,26,32,29)(34,36)(35,41)(40,44)(43,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,12)(13,15)(14,16)(17,19)(18,25)(20,26)(21,29)(22,23)(24,30)(27,28)(31,32)(33,34,38,36)(35,42,41,39)(37,40,45,44)(43,48,47,46);(3,9)(6,11)(10,15)(12,16)(17,20,24,21)(18,22,27,31)(19,26,30,29)(23,28,32,25)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(1,4)(2,8)(3,10)(5,14)(6,12)(7,13)(9,15)(11,16)(17,21,24,20)(18,22,27,31)(19,29,30,26)(23,28,32,25)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,7,2)(3,11,9,6)(4,8,13,14)(10,12,15,16)(17,22)(18,21)(19,23)(20,27)(24,31)(25,29)(26,28)(30,32)(33,36,38,34)(35,39,41,42)(37,44,45,40)(43,46,47,48);(1,6,7,11)(2,9,5,3)(4,12,13,16)(8,15,14,10)(17,23)(18,26)(19,22)(20,25)(21,28)(24,32)(27,29)(30,31)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=g,d^2=gh,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8248826);