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 | #41300 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4057 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)(19,27)(20,29)(21,26)(22,30)(23,25)(24,28)(31,32)(33,34,39,41)(35,43,45,40)(36,38,47,48)(37,42,44,46);(2,3)(4,11)(8,10)(12,13)(17,19,23,28)(18,24,25,27)(20,21,30,31)(22,26,29,32)(33,35,39,45)(34,40,41,43)(36,46,47,42)(37,38,44,48);(18,25)(19,28)(20,30)(26,32)(34,41)(36,47)(40,43)(42,46);(1,3,6,2)(4,9,11,7)(5,10,14,8)(12,16,13,15)(17,20)(18,22)(19,21)(23,30)(24,26)(25,29)(27,32)(28,31)(33,36)(34,38)(35,42)(37,40)(39,47)(41,48)(43,44)(45,46);(1,4)(2,7)(3,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,21)(18,26)(19,20)(22,24)(23,31)(25,32)(27,29)(28,30)(33,37)(34,42)(35,38)(36,40)(39,44)(41,46)(43,47)(45,48);(1,5,6,14)(2,8,3,10)(4,12,11,13)(7,15,9,16)(17,22)(18,20)(19,26)(21,24)(23,29)(25,30)(27,31)(28,32)(33,38)(34,36)(35,44)(37,45)(39,48)(40,46)(41,47)(42,43)Polycyclic presentation:
2^9|a^2=gi,b^2=h,d^2=gh,f^2=gh,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, 9845358);