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 | |
4085 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,13)(10,16)(14,15)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(1,2)(3,7)(4,13)(5,9)(6,11)(8,12)(10,14)(15,16)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(34,36)(37,42)(41,44)(45,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,14)(12,16)(13,15)(17,19,23,21)(18,25,20,24)(22,27,30,29)(26,32,28,31)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,48)(46,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,16)(13,14)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,36,39,34)(35,42,40,37)(38,41,46,44)(43,45,47,48);(1,4)(2,8)(3,10)(5,12)(6,14)(7,15)(9,13)(11,16)(17,21,23,19)(18,24,20,25)(22,29,30,27)(26,31,28,32)(33,37,39,42)(34,40,36,35)(38,45,46,48)(41,47,44,43);(1,2)(3,6)(4,8)(5,9)(7,11)(10,14)(12,13)(15,16)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8382360);