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 | |
4153 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)(3,10)(4,8)(5,9)(6,7)(11,16)(12,15)(13,14)(17,18,23,20)(19,24,22,27)(21,25,29,28)(26,31,30,32)(33,34)(35,42)(36,44)(37,41)(38,39)(40,46)(43,48)(45,47);(2,7)(3,9)(4,12)(14,16)(17,19,23,22)(18,24,20,27)(21,26,29,30)(25,31,28,32)(33,35)(34,37)(36,45)(38,42)(39,41)(40,47)(43,46)(44,48);(3,9)(4,12)(5,10)(8,15)(18,20)(19,22)(21,29)(31,32)(34,38)(36,46)(37,42)(43,45);(1,3,6,9)(2,5,7,10)(4,13,12,11)(8,16,15,14)(17,20,23,18)(19,24,22,27)(21,25,29,28)(26,32,30,31)(33,36)(34,40)(35,43)(37,48)(38,44)(39,46)(41,45)(42,47);(1,4,6,12)(2,8,7,15)(3,11,9,13)(5,14,10,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)(33,37,39,42)(34,41,38,35)(36,47,46,48)(40,43,44,45);(1,5)(2,9)(3,7)(4,14)(6,10)(8,13)(11,15)(12,16)(17,22,23,19)(18,24,20,27)(21,30,29,26)(25,31,28,32)(33,38,39,34)(35,42,41,37)(36,44,46,40)(43,47,45,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=gh,e^2=g,f^2=h,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, 8969324);