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 | |
4103 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)(3,6)(4,7)(5,10)(8,13)(9,11)(12,15)(14,16)(17,18,22,24)(19,23,20,25)(21,26,29,31)(27,30,28,32)(33,34,38,42)(35,36,40,39)(37,41,45,48)(43,44,47,46);(2,3)(7,9)(8,10)(14,15)(17,19)(18,23)(20,22)(21,27)(24,25)(26,30)(28,29)(31,32)(33,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,48)(46,47);(2,3)(4,11)(5,13)(14,15)(18,24)(19,20)(21,29)(30,32)(35,40)(36,39)(37,45)(41,48);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20,22,19)(18,25,24,23)(21,27,29,28)(26,30,31,32)(33,35)(34,39)(36,42)(37,43)(38,40)(41,46)(44,48)(45,47);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,21,22,29)(18,26,24,31)(19,27,20,28)(23,30,25,32)(33,36,38,39)(34,40,42,35)(37,44,45,46)(41,47,48,43);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,32)(29,31)(33,37,38,45)(34,41,42,48)(35,43,40,47)(36,44,39,46)Polycyclic presentation:
2^9|a^2=gh,d^2=h,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, 8966349);