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 | #46871 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4095 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,10)(3,8,14,9)(4,15,5,6)(11,13,12,16)(17,18)(19,26)(20,28)(21,23)(22,24)(25,30)(27,31)(29,32)(33,34,39,38)(35,42,40,36)(37,45,46,41)(43,44,47,48);(1,3,7,14)(2,8,10,9)(4,12,5,11)(6,13,15,16)(17,19,23,22)(18,24,21,26)(20,29,30,27)(25,32,28,31)(33,35,39,40)(34,36,38,42)(37,43,46,47)(41,44,45,48);(1,4,7,5)(2,6,10,15)(3,11,14,12)(8,16,9,13)(17,18,23,21)(19,24,22,26)(20,25,30,28)(27,31,29,32)(33,36,39,42)(34,40,38,35)(37,44,46,48)(41,47,45,43);(1,5,7,4)(2,6,10,15)(3,12,14,11)(8,16,9,13)(17,20)(18,25)(19,27)(21,28)(22,29)(23,30)(24,31)(26,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,3)(2,9)(4,11)(5,12)(6,13)(7,14)(8,10)(15,16)(17,21,23,18)(19,24,22,26)(20,28,30,25)(27,31,29,32)(33,37,39,46)(34,41,38,45)(35,43,40,47)(36,44,42,48);(1,6)(2,4)(3,13)(5,10)(7,15)(8,12)(9,11)(14,16)(17,22,23,19)(18,24,21,26)(20,29,30,27)(25,31,28,32)(33,38,39,34)(35,42,40,36)(37,45,46,41)(43,48,47,44)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8109673);