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 | |
4164 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18,22,23)(19,20,24,27)(21,25,30,31)(26,28,29,32)(33,34)(35,39)(36,37)(38,42)(40,45)(41,48)(43,44)(46,47);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(18,23)(20,27)(25,31)(28,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,42)(41,46)(47,48);(3,11)(4,12)(6,16)(8,13)(17,19,22,24)(18,20,23,27)(21,29,30,26)(25,32,31,28)(33,35,38,45)(34,40,42,39)(36,43,41,46)(37,47,48,44);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,19)(18,20)(21,26)(22,24)(23,27)(25,28)(29,30)(31,32)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,20)(18,24)(19,23)(21,28)(22,27)(25,29)(26,31)(30,32)(33,36,38,41)(34,37,42,48)(35,43,45,46)(39,44,40,47);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,21,22,30)(18,25,23,31)(19,26,24,29)(20,28,27,32)(33,37)(34,41)(35,44)(36,42)(38,48)(39,46)(40,43)(45,47)Polycyclic presentation:
2^9|a^2=g,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,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, 8834912);