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 | |
4133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31)(33,34,39,42)(35,40,36,37)(38,41,47,48)(43,46,44,45);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(19,25)(22,29)(26,31)(30,32)(33,35,39,36)(34,37,42,40)(38,43,47,44)(41,45,48,46);(2,9)(3,4)(10,16)(11,12)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32)(33,35,39,36)(34,40,42,37)(38,43,47,44)(41,46,48,45);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36,39,35)(34,37,42,40)(38,44,47,43)(41,45,48,46);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,37)(34,35)(36,42)(38,46)(39,40)(41,44)(43,48)(45,47);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,46)(42,48)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9818108);