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 | |
4081 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)(3,12)(4,6)(5,8)(7,9)(10,11)(13,16)(14,15)(17,18)(19,22)(20,27)(21,24)(23,28)(25,31)(26,30)(29,32)(33,34,39,42)(35,40,46,37)(36,38,48,41)(43,47,44,45);(1,3,7,5)(2,8,9,12)(4,13,11,15)(6,14,10,16)(17,19,21,26)(18,22,24,30)(20,25,28,32)(23,29,27,31)(33,35,39,46)(34,40,42,37)(36,44,48,43)(38,45,41,47);(2,9)(4,11)(8,12)(13,15)(17,18,21,24)(19,22,26,30)(20,23,28,27)(25,29,32,31)(33,36,39,48)(34,41,42,38)(35,43,46,44)(37,47,40,45);(1,4,7,11)(2,10,9,6)(3,13,5,15)(8,16,12,14)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,36)(34,38)(35,44)(37,47)(39,48)(40,45)(41,42)(43,46);(1,5,7,3)(2,8,9,12)(4,15,11,13)(6,14,10,16)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,31)(28,32)(33,37)(34,35)(36,47)(38,44)(39,40)(41,43)(42,46)(45,48);(1,6,7,10)(2,11,9,4)(3,14,5,16)(8,15,12,13)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31)(33,38)(34,36)(35,45)(37,43)(39,41)(40,44)(42,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9690183);