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 | |
4054 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,11)(8,13)(9,15)(14,16)(17,18)(19,26)(20,28)(21,23)(22,24)(25,30)(27,31)(29,32)(34,40)(36,38)(41,48)(44,46);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,19,23,22)(18,24,21,26)(20,29,30,27)(25,32,28,31)(33,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,46)(47,48);(2,6)(7,12)(8,13)(14,16)(18,21)(24,26)(25,28)(31,32)(33,35,39,42)(34,36,40,38)(37,45,47,43)(41,46,48,44);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,20,23,30)(18,25,21,28)(19,27,22,29)(24,31,26,32)(33,36,39,38)(34,35,40,42)(37,44,47,46)(41,43,48,45);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,21,23,18)(19,26,22,24)(20,28,30,25)(27,32,29,31)(33,37,39,47)(34,41,40,48)(35,43,42,45)(36,44,38,46);(1,4)(2,8)(3,9)(5,11)(6,13)(7,14)(10,15)(12,16)(17,22,23,19)(18,26,21,24)(20,29,30,27)(25,32,28,31)(33,38,39,36)(34,42,40,35)(37,46,47,44)(41,45,48,43)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8025157);