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 | |
4135 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,14)(13,16)(19,25)(23,26)(27,31)(30,32)(33,34)(35,42)(36,40)(37,41)(38,39)(43,44)(45,46)(47,48);(2,7)(4,13)(8,14)(10,11)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,35)(34,37)(36,43)(38,42)(39,41)(40,47)(44,45)(46,48);(4,13)(5,15)(9,16)(10,11)(18,22)(20,28)(23,26)(27,31)(34,38)(37,42)(40,45)(44,47);(1,3)(2,8)(4,10)(5,9)(6,12)(7,14)(11,13)(15,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,39,46)(34,40,38,45)(35,43,41,48)(37,47,42,44);(1,4)(2,9)(3,11)(5,14)(6,13)(7,16)(8,15)(10,12)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,37,39,42)(34,41,38,35)(36,44,46,47)(40,43,45,48);(1,5,6,15)(2,10,7,11)(3,9,12,16)(4,14,13,8)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,38,39,34)(35,42,41,37)(36,45,46,40)(43,44,48,47)Polycyclic presentation:
2^9|b^2=i,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,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, 8946649);