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 | |
4110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,9)(3,8,4,14)(5,10,15,13)(6,12,16,11)(18,22)(20,28)(23,25)(26,32)(33,34,39,41)(35,42,46,37)(36,38,48,40)(43,47,44,45);(1,3)(2,8)(4,7)(5,12)(6,13)(9,14)(10,16)(11,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35)(34,37)(36,43)(38,45)(39,46)(40,47)(41,42)(44,48);(2,9)(3,4)(10,13)(11,12)(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,36,39,48)(34,40,41,38)(35,43,46,44)(37,47,42,45);(1,4,7,3)(2,8,9,14)(5,11,15,12)(6,10,16,13)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36)(34,38)(35,44)(37,47)(39,48)(40,41)(42,45)(43,46);(1,5)(2,10)(3,12)(4,11)(6,14)(7,15)(8,16)(9,13)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,37)(34,35)(36,47)(38,44)(39,42)(40,43)(41,46)(45,48);(1,6,7,16)(2,11,9,12)(3,13,4,10)(5,8,15,14)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,36)(35,45)(37,43)(39,40)(41,48)(42,44)(46,47)Polycyclic presentation:
2^9|a^2=gi,c^2=g,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9845273);