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 | |
4300 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)(3,4)(5,9)(6,7)(8,11)(10,12)(13,16)(14,15)(17,18,22,24)(19,23,27,30)(20,21,28,29)(25,26,31,32)(34,39)(35,42)(37,45)(44,47);(2,7)(4,8)(9,15)(12,13)(17,19,22,27)(18,23,24,30)(20,25,28,31)(21,26,29,32)(33,34,38,39)(35,36,42,40)(37,41,45,46)(43,44,48,47);(18,24)(20,28)(23,30)(25,31)(33,35,38,42)(34,40,39,36)(37,43,45,48)(41,47,46,44);(1,3)(2,8)(4,7)(5,10)(6,11)(9,13)(12,15)(14,16)(17,20)(18,21)(19,25)(22,28)(23,26)(24,29)(27,31)(30,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,4,6,8)(2,3,7,11)(5,13,14,12)(9,16,15,10)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,5,6,14)(2,9,7,15)(3,10,11,16)(4,12,8,13)(17,21)(18,20)(19,26)(22,29)(23,25)(24,28)(27,32)(30,31)(33,37,38,45)(34,41,39,46)(35,43,42,48)(36,44,40,47)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9757823);