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 | |
4068 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)(3,8)(4,5)(6,14)(7,9)(10,15)(11,13)(12,16)(19,25)(23,30)(26,28)(31,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(1,3,7,4)(2,8,9,5)(6,11,15,12)(10,16,14,13)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,34,38,42)(35,39,40,36)(37,41,46,48)(43,47,45,44);(2,9)(3,4)(6,15)(13,16)(18,22)(19,25)(24,27)(26,28)(33,35,38,40)(34,39,42,36)(37,45,46,43)(41,44,48,47);(1,4,7,3)(2,8,9,5)(6,12,15,11)(10,16,14,13)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,35)(34,39)(36,42)(37,43)(38,40)(41,47)(44,48)(45,46);(1,5,7,8)(2,4,9,3)(6,13,15,16)(10,12,14,11)(17,20,21,29)(18,24,22,27)(19,26,25,28)(23,31,30,32)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,45,48,43);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,16)(9,14)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,45)(42,48)Polycyclic presentation:
2^9|b^2=gh,c^2=g,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8838248);