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 | |
4113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(3,9)(7,14)(10,12)(15,16)(17,18,22,24)(19,23,27,32)(20,25,28,29)(21,30,31,26)(33,34)(35,37)(36,40)(38,42)(39,41)(43,45)(44,46)(47,48);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,19,22,27)(18,23,24,32)(20,21,28,31)(25,30,29,26)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,47,44,45);(2,6)(3,9)(8,11)(10,12)(18,24)(19,27)(21,31)(25,29)(33,34,38,42)(35,39,41,37)(36,40,46,44)(43,47,48,45);(1,3,5,9)(2,7,6,14)(4,12,13,10)(8,16,11,15)(17,18)(19,23)(20,29)(21,26)(22,24)(25,28)(27,32)(30,31)(33,34)(35,39)(36,44)(37,41)(38,42)(40,46)(43,45)(47,48);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,20,22,28)(18,25,24,29)(19,21,27,31)(23,30,32,26)(33,36,38,46)(34,40,42,44)(35,43,41,48)(37,45,39,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,21)(18,26)(19,28)(20,27)(22,31)(23,25)(24,30)(29,32)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,48,44,43)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=ghi,e^2=h,f^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, 8837519);