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 | |
4156 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)(3,10)(4,14)(5,12)(6,8)(7,13)(9,15)(11,16)(17,18,23,25)(19,29,30,24)(20,21,27,26)(22,32,31,28)(33,34)(35,40)(36,37)(38,47)(39,43)(41,46)(42,44)(45,48);(1,3)(2,7)(4,12)(5,9)(6,13)(8,10)(11,15)(14,16)(17,19)(18,24)(20,31)(21,32)(22,27)(23,30)(25,29)(26,28)(33,35,39,45)(34,40,43,48)(36,44,46,38)(37,42,41,47);(2,8)(4,15)(7,10)(11,12)(18,25)(19,30)(20,27)(28,32)(33,36,39,46)(34,41,43,37)(35,38,45,44)(40,42,48,47);(1,4)(2,9)(3,12)(5,7)(6,15)(8,14)(10,16)(11,13)(17,20,23,27)(18,26,25,21)(19,31,30,22)(24,28,29,32)(33,36)(34,37)(35,44)(38,45)(39,46)(40,42)(41,43)(47,48);(1,3)(2,10)(4,12)(5,14)(6,13)(7,8)(9,16)(11,15)(17,21,23,26)(18,27,25,20)(19,28,30,32)(22,24,31,29)(33,37)(34,36)(35,42)(38,48)(39,41)(40,44)(43,46)(45,47);(1,5,6,16)(2,11,8,12)(3,9,13,14)(4,7,15,10)(17,22,23,31)(18,28,25,32)(19,27,30,20)(21,24,26,29)(33,38,39,44)(34,42,43,47)(35,36,45,46)(37,40,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9770624);