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 | |
4045 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,14)(15,16)(17,18,22,24)(19,25,28,23)(20,29,31,21)(26,32,30,27)(33,34,39,43)(35,40,44,48)(36,37,41,47)(38,42,46,45);(2,7)(4,8)(9,14)(12,15)(17,19,22,28)(18,23,24,25)(20,30,31,26)(21,27,29,32)(33,35)(34,40)(36,45)(37,38)(39,44)(41,42)(43,48)(46,47);(5,13)(9,14)(10,16)(12,15)(18,24)(21,29)(23,25)(27,32)(35,44)(38,46)(40,48)(42,45);(1,3,6,11)(2,8,7,4)(5,10,13,16)(9,15,14,12)(17,19)(18,23)(20,30)(21,27)(22,28)(24,25)(26,31)(29,32)(33,36)(34,37)(35,45)(38,40)(39,41)(42,44)(43,47)(46,48);(1,4,6,8)(2,3,7,11)(5,12,13,15)(9,10,14,16)(17,20)(18,21)(19,26)(22,31)(23,32)(24,29)(25,27)(28,30)(33,37)(34,41)(35,46)(36,43)(38,44)(39,47)(40,45)(42,48);(1,5,6,13)(2,9,7,14)(3,10,11,16)(4,12,8,15)(17,21)(18,20)(19,27)(22,29)(23,30)(24,31)(25,26)(28,32)(33,38,39,46)(34,42,43,45)(35,37,44,47)(36,40,41,48)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=gh,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8281411);