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 | |
4155 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31)(33,34,38,42)(35,36,40,39)(37,45,46,41)(43,47,48,44);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(18,20)(19,26)(24,28)(27,32)(33,35,38,40)(34,39,42,36)(37,43,46,48)(41,47,45,44);(2,6)(7,9)(8,13)(14,15)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,48,45,43);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,48)(46,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,20,22,18)(19,25,26,23)(21,28,29,24)(27,31,32,30)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,47)(46,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31)(33,37,38,46)(34,41,42,45)(35,43,40,48)(36,44,39,47)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000656);