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 | |
3991 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,7,6)(3,11,14,8)(4,9,13,15)(5,16,12,10)(19,25)(23,26)(27,29)(31,32)(33,34)(35,40)(36,44)(37,42)(38,39)(41,46)(43,45)(47,48);(1,3,7,14)(2,8,6,11)(4,12,13,5)(9,16,15,10)(17,18)(19,26)(20,28)(21,22)(23,25)(24,30)(27,31)(29,32)(34,39)(35,42)(41,44)(43,48);(2,6)(8,11)(9,15)(10,16)(18,22)(23,26)(24,28)(31,32)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,4)(2,9)(3,12)(5,14)(6,15)(7,13)(8,16)(10,11)(17,19,21,25)(18,23,22,26)(20,29,30,27)(24,32,28,31)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,5,7,12)(2,10,6,16)(3,13,14,4)(8,15,11,9)(17,20,21,30)(18,24,22,28)(19,27,25,29)(23,31,26,32);(1,6,7,2)(3,11,14,8)(4,15,13,9)(5,16,12,10)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8323958);