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 | |
4112 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)(3,7)(4,9)(5,10)(6,8)(11,12)(13,15)(14,16)(17,18,23,25)(19,28,22,24)(20,26,30,32)(21,27,31,29)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(1,3,6,10)(2,7,8,5)(4,11,13,16)(9,12,15,14)(17,19,23,22)(18,24,25,28)(20,29,30,27)(21,26,31,32)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(2,8)(3,10)(9,15)(11,16)(18,25)(20,30)(24,28)(27,29)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(1,3)(2,5)(4,11)(6,10)(7,8)(9,14)(12,15)(13,16)(17,20)(18,26)(19,27)(21,24)(22,29)(23,30)(25,32)(28,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,21)(18,27)(19,26)(20,24)(22,32)(23,31)(25,29)(28,30)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,5)(2,3)(4,12)(6,7)(8,10)(9,16)(11,15)(13,14)(17,22,23,19)(18,28,25,24)(20,29,30,27)(21,26,31,32)(33,36)(34,39)(35,41)(37,44)(38,45)(40,43)(42,48)(46,47)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9846766);