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 | |
4234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(4,12)(8,15)(10,16)(13,14)(17,18,23,26)(19,24,21,20)(22,25,31,30)(27,32,29,28)(33,34,40,38)(35,44,46,41)(36,47,42,37)(39,43,45,48);(1,2)(3,7)(4,8)(5,9)(6,11)(10,14)(12,15)(13,16)(17,19,23,21)(18,20,26,24)(22,27,31,29)(25,28,30,32)(33,35)(34,41)(36,45)(37,48)(38,44)(39,42)(40,46)(43,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,20,23,24)(18,19,26,21)(22,28,31,32)(25,27,30,29)(33,36)(34,37)(35,45)(38,47)(39,46)(40,42)(41,48)(43,44);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,18)(19,20)(21,24)(22,30)(23,26)(25,31)(27,32)(28,29)(33,37,40,47)(34,42,38,36)(35,43,46,48)(39,41,45,44);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,23,19)(18,24,26,20)(22,29,31,27)(25,32,30,28)(33,38,40,34)(35,44,46,41)(36,47,42,37)(39,48,45,43);(1,2)(3,6)(4,8)(5,9)(7,11)(10,13)(12,15)(14,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,39)(34,43)(35,42)(36,46)(37,44)(38,48)(40,45)(41,47)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8337692);