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 | #41349 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,3)(4,7,11,9)(5,8,14,10)(12,15,13,16)(17,18)(19,21)(20,24)(22,23)(25,29)(26,28)(27,30)(31,32)(33,34,39,40)(35,38,36,43)(37,41,46,48)(42,47,44,45);(2,3)(4,11)(8,10)(12,13)(20,27)(24,30)(26,31)(28,32)(33,35)(34,38)(36,39)(37,42)(40,43)(41,47)(44,46)(45,48);(18,23)(19,25)(24,30)(26,31)(34,40)(35,36)(41,48)(42,44);(1,3,6,2)(4,9,11,7)(5,10,14,8)(12,16,13,15)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,36,39,35)(34,38,40,43)(37,44,46,42)(41,47,48,45);(1,4)(2,7)(3,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,37)(34,41)(35,42)(36,44)(38,47)(39,46)(40,48)(43,45);(1,5,6,14)(2,8,3,10)(4,12,11,13)(7,15,9,16)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,38)(34,35)(36,40)(37,45)(39,43)(41,44)(42,48)(46,47)Polycyclic presentation:
2^9|a^2=gi,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 9843574);