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 | |
4166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,10)(11,15)(14,16)(17,18)(19,25)(20,31)(21,22)(23,26)(24,30)(27,29)(28,32)(35,36)(38,42)(43,44)(45,47);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,26)(31,32)(33,34,39,41)(35,42,36,38)(37,40,46,48)(43,47,44,45);(2,6)(4,12)(7,10)(11,15)(17,20,23,21)(18,22,26,31)(19,27,30,28)(24,29,25,32)(33,35,39,36)(34,38,41,42)(37,44,46,43)(40,47,48,45);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,21,23,20)(18,22,26,31)(19,28,30,27)(24,29,25,32)(33,36,39,35)(34,38,41,42)(37,43,46,44)(40,47,48,45);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19)(18,25)(20,28)(21,27)(22,29)(23,30)(24,26)(31,32)(33,37,39,46)(34,40,41,48)(35,43,36,44)(38,45,42,47);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,22)(18,20)(19,29)(21,26)(23,31)(24,27)(25,28)(30,32)(33,38)(34,35)(36,41)(37,45)(39,42)(40,43)(44,48)(46,47)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,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, 8946567);