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 | |
3977 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)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18,22,23)(19,26,20,24)(21,25,29,30)(27,32,28,31)(33,34)(35,37)(36,44)(38,39)(40,42)(41,46)(43,47)(45,48);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(18,23)(24,26)(25,30)(31,32)(34,39)(35,42)(41,44)(43,48);(4,12)(7,14)(10,16)(13,15)(17,19,22,20)(18,24,23,26)(21,28,29,27)(25,32,30,31)(33,34,38,39)(35,40,42,37)(36,41,46,44)(43,47,48,45);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,20,22,19)(18,24,23,26)(21,28,29,27)(25,31,30,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,24)(20,26)(21,25)(22,23)(27,31)(28,32)(29,30)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,21,22,29)(18,25,23,30)(19,27,20,28)(24,31,26,32)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=h,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8324231);