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 | |
4311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,8,14,10)(12,15,13,16)(19,25)(23,26)(27,31)(30,32)(34,38)(36,44)(39,42)(43,45);(2,3)(7,9)(8,10)(15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(34,38)(35,41)(40,46)(43,45);(4,11)(7,9)(12,13)(15,16)(18,22)(23,26)(24,28)(30,32)(33,34,37,38)(35,42,41,39)(36,40,44,46)(43,48,45,47);(1,3,6,2)(4,9,11,7)(5,10,14,8)(12,16,13,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,46)(45,48);(1,4,6,11)(2,7,3,9)(5,13,14,12)(8,16,10,15)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31)(33,35)(34,39)(36,45)(37,41)(38,42)(40,48)(43,44)(46,47);(1,5,6,14)(2,8,3,10)(4,12,11,13)(7,15,9,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,36)(34,40)(35,43)(37,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9927839);