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 | |
4079 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)(17,18,22,23)(19,20,25,24)(21,26,30,29)(27,28,32,31)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(1,2)(3,7)(4,13)(5,9)(6,11)(8,12)(10,14)(15,16)(18,23)(20,24)(26,29)(28,31)(33,35,38,43)(34,39,37,41)(36,42,45,48)(40,46,44,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,16)(13,15)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,31,29,28)(33,34,38,37)(35,39,43,41)(36,40,45,44)(42,46,48,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,16)(13,14)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,28)(31,32)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48);(1,4)(2,8)(3,10)(5,12)(6,14)(7,15)(9,13)(11,16)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,21,22,30)(18,26,23,29)(19,27,25,32)(20,28,24,31)(33,37,38,34)(35,41,43,39)(36,40,45,44)(42,46,48,47)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,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8379567);