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 | |
4308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,5,4,11)(6,14,15,9)(10,16,12,13)(17,18,22,20)(19,24,27,23)(21,28,29,25)(26,30,32,31)(33,34)(35,38)(36,39)(37,40)(41,42)(43,45)(44,46)(47,48);(1,3,7,4)(2,5,8,11)(6,10,15,12)(9,13,14,16)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,34,37,40)(35,38,41,42)(36,39,44,46)(43,45,47,48);(2,8)(3,4)(6,15)(13,16)(18,20)(21,29)(23,24)(26,32)(35,41)(36,44)(38,42)(39,46);(1,4,7,3)(2,5,8,11)(6,12,15,10)(9,13,14,16)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,35)(34,38)(36,43)(37,41)(39,45)(40,42)(44,47)(46,48);(1,5)(2,4)(3,8)(6,13)(7,11)(9,12)(10,14)(15,16)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(1,6)(2,9)(3,10)(4,12)(5,13)(7,15)(8,14)(11,16)(17,21,22,29)(18,25,20,28)(19,26,27,32)(23,30,24,31)(33,36,37,44)(34,39,40,46)(35,43,41,47)(38,45,42,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=h,f^2=g,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, 8834554);