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 | #35405 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4059 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,8)(4,12)(5,11)(6,7)(9,15)(10,14)(13,16)(19,25)(20,28)(23,26)(24,29)(33,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,47)(46,48);(1,3,6,11)(2,5,7,8)(4,10,15,13)(9,14,12,16)(17,18)(19,26)(20,29)(21,22)(23,25)(24,28)(27,31)(30,32)(33,35,38,40)(34,39,42,36)(37,43,46,45)(41,47,48,44);(2,7)(5,8)(9,12)(14,16)(18,22)(23,26)(24,29)(31,32)(33,35,38,40)(34,36,42,39)(37,45,46,43)(41,47,48,44);(1,3)(2,8)(4,13)(5,7)(6,11)(9,14)(10,15)(12,16)(17,19)(18,23)(20,30)(21,25)(22,26)(24,32)(27,28)(29,31)(33,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,47)(46,48);(1,4)(2,9)(3,10)(5,14)(6,15)(7,12)(8,16)(11,13)(17,20)(18,24)(19,27)(21,28)(22,29)(23,31)(25,30)(26,32)(33,36)(34,40)(35,42)(37,44)(38,39)(41,45)(43,48)(46,47);(1,5,6,8)(2,3,7,11)(4,14,15,16)(9,10,12,13)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,37,38,46)(34,41,42,48)(35,43,40,45)(36,44,39,47)Polycyclic presentation:
2^9|b^2=g,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8839912);