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 | |
4127 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,8)(3,9,4,5)(6,10,14,15)(11,16,12,13)(17,18)(19,26)(20,30)(21,24)(22,31)(23,25)(27,32)(28,29)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,46)(47,48);(2,8)(5,9)(10,15)(13,16)(17,19,23,21)(18,24,25,26)(20,28,31,27)(22,29,30,32)(33,35)(34,36)(37,43)(38,39)(40,44)(41,42)(45,48)(46,47);(1,3,7,4)(2,9,8,5)(6,11,14,12)(10,16,15,13)(17,20)(18,22)(19,27)(21,28)(23,31)(24,32)(25,30)(26,29)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,4,7,3)(2,9,8,5)(6,11,14,12)(10,13,15,16)(17,20)(18,22)(19,28)(21,27)(23,31)(24,29)(25,30)(26,32)(33,34)(35,36)(37,45)(38,41)(39,42)(40,46)(43,48)(44,47);(1,5,7,9)(2,4,8,3)(6,13,14,16)(10,12,15,11)(17,21,23,19)(18,24,25,26)(20,28,31,27)(22,32,30,29)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,6,7,14)(2,10,8,15)(3,11,4,12)(5,13,9,16)(17,22)(18,20)(19,29)(21,32)(23,30)(24,28)(25,31)(26,27)(33,37,38,46)(34,40,41,45)(35,43,39,47)(36,44,42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=gh,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10353642);