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 | |
3896 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)(3,10)(4,9)(5,14)(6,8)(7,13)(11,12)(15,16)(17,18,23,26)(19,24,20,21)(22,25,31,32)(27,30,28,29)(35,41)(39,45)(42,47)(46,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,10)(9,15)(14,16)(17,19,23,20)(18,21,26,24)(22,27,31,28)(25,29,32,30)(34,38)(39,45)(40,43)(46,48);(2,8)(4,14)(7,10)(11,16)(17,19,23,20)(18,24,26,21)(22,27,31,28)(25,30,32,29)(33,34)(35,39)(36,40)(37,38)(41,45)(42,46)(43,44)(47,48);(1,4)(2,9)(3,11)(5,8)(6,14)(7,15)(10,12)(13,16)(17,20,23,19)(18,21,26,24)(22,28,31,27)(25,29,32,30)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47);(1,3)(2,10)(4,11)(5,15)(6,13)(7,8)(9,12)(14,16)(17,21)(18,19)(20,26)(22,30)(23,24)(25,28)(27,32)(29,31)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47);(1,5,6,9)(2,4,8,14)(3,12,13,15)(7,11,10,16)(17,22,23,31)(18,25,26,32)(19,27,20,28)(21,29,24,30)(33,36)(34,40)(35,42)(37,44)(38,43)(39,46)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,f^2=g,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, 9817894);