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 | |
4141 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)(3,8)(4,10)(5,11)(6,14)(7,9)(12,16)(13,15)(34,39)(35,41)(37,45)(44,48);(1,3)(2,8)(4,7)(5,12)(6,13)(9,10)(11,16)(14,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,31)(30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,46)(45,47);(2,9)(3,4)(5,15)(6,16)(18,22)(19,26)(20,29)(31,32)(33,35,38,41)(34,40,39,36)(37,46,45,43)(42,44,47,48);(1,4,7,3)(2,10,9,8)(5,12,15,14)(6,11,16,13)(17,19)(18,23)(20,30)(21,26)(22,25)(24,32)(27,29)(28,31)(33,35)(34,36)(37,43)(38,41)(39,40)(42,44)(45,46)(47,48);(1,5,7,15)(2,11,9,13)(3,12,4,14)(6,8,16,10)(17,20)(18,24)(19,27)(21,29)(22,28)(23,31)(25,32)(26,30)(33,36)(34,41)(35,39)(37,44)(38,40)(42,46)(43,47)(45,48);(1,6)(2,12)(3,13)(4,11)(5,8)(7,16)(9,14)(10,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,37,38,45)(34,42,39,47)(35,43,41,46)(36,44,40,48)Polycyclic presentation:
2^9|c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8966308);