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 | |
3913 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,31)(30,32)(33,34,39,42)(35,37,41,38)(36,44,47,40)(43,46,48,45);(3,9)(7,13)(10,15)(14,16)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(33,35,39,41)(34,38,42,37)(36,43,47,48)(40,46,44,45);(2,6)(7,13)(8,11)(14,16)(21,29)(24,31)(26,32)(28,30)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,46)(47,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,32)(29,31)(33,36,39,47)(34,40,42,44)(35,43,41,48)(37,45,38,46);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,37,39,38)(34,41,42,35)(36,45,47,46)(40,48,44,43);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,38,39,37)(34,35,42,41)(36,46,47,45)(40,43,44,48)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388669);