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 | |
4122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,5,10,8)(4,9,12,15)(11,14,13,16)(17,18)(19,27)(20,25)(21,29)(22,23)(24,28)(26,31)(30,32)(33,34,38,40)(35,39,36,41)(37,42,45,47)(43,46,44,48);(2,7)(3,10)(4,12)(14,16)(18,23)(19,28)(25,32)(26,29)(33,35)(34,39)(36,38)(37,43)(40,41)(42,46)(44,45)(47,48);(3,10)(4,12)(5,8)(9,15)(18,23)(19,28)(20,30)(21,31)(34,40)(35,36)(37,45)(46,48);(1,3,6,10)(2,8,7,5)(4,13,12,11)(9,14,15,16)(17,19)(18,24)(20,26)(21,25)(22,28)(23,27)(29,30)(31,32)(33,36,38,35)(34,41,40,39)(37,43,45,44)(42,46,47,48);(1,4,6,12)(2,9,7,15)(3,11,10,13)(5,14,8,16)(17,20)(18,25)(19,29)(21,27)(22,30)(23,32)(24,31)(26,28)(33,37,38,45)(34,42,40,47)(35,43,36,44)(39,46,41,48);(1,5,6,8)(2,10,7,3)(4,14,12,16)(9,13,15,11)(17,21,22,31)(18,26,23,29)(19,25,28,32)(20,27,30,24)(33,34)(35,39)(36,41)(37,42)(38,40)(43,46)(44,48)(45,47)Polycyclic presentation:
2^9|a^2=gh,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, 8966357);