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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,9,11)(5,8,13,12)(10,14,15,16)(17,18,23,24)(19,21,20,27)(22,30,31,25)(26,32,28,29)(33,34)(35,42)(36,39)(37,45)(38,40)(41,46)(43,47)(44,48);(2,4)(7,11)(8,12)(14,16)(17,19,23,20)(18,21,24,27)(22,26,31,28)(25,29,30,32)(33,35,38,36)(34,39,40,42)(37,43,46,44)(41,47,45,48);(3,9)(5,13)(7,11)(8,12)(18,24)(19,20)(22,31)(29,32)(34,40)(37,46)(39,42)(43,44);(1,3,6,9)(2,7,4,11)(5,10,13,15)(8,14,12,16)(17,20,23,19)(18,21,24,27)(22,28,31,26)(25,29,30,32)(33,34)(35,39)(36,42)(37,41)(38,40)(43,47)(44,48)(45,46);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,21)(18,20)(19,24)(22,29)(23,27)(25,28)(26,30)(31,32)(33,36,38,35)(34,39,40,42)(37,44,46,43)(41,47,45,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,22)(18,25)(19,26)(20,28)(21,29)(23,31)(24,30)(27,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,45)(42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8834545);