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 | #48366 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3943 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)(3,12)(4,9)(5,10)(6,8)(7,11)(13,16)(14,15)(17,18,22,26)(19,23,28,32)(20,29,30,24)(21,25,27,31)(33,34,40,37)(35,44,46,41)(36,47,38,42)(39,48,45,43);(1,3,7,6)(2,8,11,12)(4,13,5,14)(9,15,10,16)(17,19,22,28)(18,23,26,32)(20,21,30,27)(24,31,29,25)(33,35)(34,41)(36,45)(37,44)(38,39)(40,46)(42,43)(47,48);(1,4,7,5)(2,9,11,10)(3,13,6,14)(8,15,12,16)(17,20,22,30)(18,24,26,29)(19,27,28,21)(23,25,32,31)(33,36,40,38)(34,42,37,47)(35,45,46,39)(41,43,44,48);(1,5,7,4)(2,10,11,9)(3,14,6,13)(8,16,12,15)(17,20)(18,24)(19,21)(22,30)(23,31)(25,32)(26,29)(27,28)(33,37,40,34)(35,44,46,41)(36,42,38,47)(39,48,45,43);(1,6,7,3)(2,8,11,12)(4,14,5,13)(9,15,10,16)(17,21)(18,25)(19,20)(22,27)(23,29)(24,32)(26,31)(28,30)(33,38,40,36)(34,42,37,47)(35,45,46,39)(41,48,44,43);(1,2)(3,8)(4,9)(5,10)(6,12)(7,11)(13,15)(14,16)(17,18)(19,23)(20,24)(21,31)(22,26)(25,27)(28,32)(29,30)(33,39,40,45)(34,43,37,48)(35,38,46,36)(41,42,44,47)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9630354);