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 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3777 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18,23,24)(19,21,26,22)(20,28,31,25)(27,30,32,29)(33,34,39,38)(35,41,36,40)(37,45,46,42)(43,47,44,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(18,24)(19,26)(25,28)(27,32)(33,35,39,36)(34,40,38,41)(37,44,46,43)(42,48,45,47);(2,6)(7,9)(8,13)(14,15)(17,19)(18,21)(20,27)(22,24)(23,26)(25,29)(28,30)(31,32)(33,36,39,35)(34,41,38,40)(37,44,46,43)(42,48,45,47);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,23,31)(18,25,24,28)(19,27,26,32)(21,29,22,30)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,21,23,22)(18,26,24,19)(20,29,31,30)(25,32,28,27)(33,34,39,38)(35,40,36,41)(37,42,46,45)(43,47,44,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,22,23,21)(18,19,24,26)(20,30,31,29)(25,27,28,32)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000800);