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 | |
3998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(8,15)(10,13)(11,16)(17,18,23,28)(19,27,30,24)(20,25,31,32)(21,26,29,22)(33,34,39,37)(35,42,41,38)(36,44,46,40)(43,45,47,48);(1,2,6,7)(3,8,9,15)(4,12,14,5)(10,11,13,16)(17,19,23,30)(18,24,28,27)(20,29,31,21)(22,25,26,32)(34,37)(35,41)(40,44)(43,47);(2,7)(5,12)(8,15)(11,16)(17,20,23,31)(18,25,28,32)(19,29,30,21)(22,24,26,27)(33,35)(34,38)(36,43)(37,42)(39,41)(40,45)(44,48)(46,47);(1,3)(2,8)(4,13)(5,11)(6,9)(7,15)(10,14)(12,16)(17,21)(18,26)(19,20)(22,28)(23,29)(24,32)(25,27)(30,31)(33,36)(34,40)(35,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,4)(2,5)(3,10)(6,14)(7,12)(8,16)(9,13)(11,15)(17,19)(18,27)(20,29)(21,31)(22,25)(23,30)(24,28)(26,32)(33,37,39,34)(35,42,41,38)(36,44,46,40)(43,48,47,45);(1,5)(2,4)(3,11)(6,12)(7,14)(8,13)(9,16)(10,15)(17,22,23,26)(18,21,28,29)(19,25,30,32)(20,27,31,24)(33,38,39,42)(34,41,37,35)(36,45,46,48)(40,47,44,43)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833548);