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 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3827 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,5,9,15)(4,6,14,8)(11,16,12,13)(17,18)(19,26)(20,29)(21,23)(22,25)(24,28)(27,31)(30,32)(33,34)(35,42)(36,41)(37,40)(38,43)(39,44)(45,47)(46,48);(1,3,7,9)(2,5,10,15)(4,12,14,11)(6,13,8,16)(17,19)(18,24)(20,27)(21,26)(22,30)(23,28)(25,31)(29,32)(33,35,39,37)(34,40,44,42)(36,45,46,38)(41,43,48,47);(1,4,7,14)(2,8,10,6)(3,11,9,12)(5,13,15,16)(17,20,23,22)(18,25,21,29)(19,27,28,30)(24,31,26,32)(33,36,39,46)(34,41,44,48)(35,45,37,38)(40,43,42,47);(1,4)(2,6)(3,12)(5,13)(7,14)(8,10)(9,11)(15,16)(17,21,23,18)(19,26,28,24)(20,25,22,29)(27,31,30,32)(33,37,39,35)(34,42,44,40)(36,38,46,45)(41,47,48,43);(1,5,7,15)(2,9,10,3)(4,13,14,16)(6,11,8,12)(17,22,23,20)(18,25,21,29)(19,27,28,30)(24,32,26,31)(33,38)(34,43)(35,36)(37,46)(39,45)(40,48)(41,42)(44,47);(1,6)(2,4)(3,13)(5,12)(7,8)(9,16)(10,14)(11,15)(17,19)(18,24)(20,27)(21,26)(22,30)(23,28)(25,31)(29,32)(33,34)(35,40)(36,41)(37,42)(38,47)(39,44)(43,45)(46,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9592698);