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 | |
4166 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)(3,4)(5,8)(6,9)(7,11)(10,12)(13,15)(14,16)(19,25)(23,30)(26,28)(31,32)(33,34,40,38)(35,43,42,39)(36,46,37,41)(44,48,45,47);(1,3,6,11)(2,7,9,4)(5,10,13,16)(8,14,15,12)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35)(34,39)(36,44)(37,45)(38,43)(40,42)(41,48)(46,47);(5,13)(8,15)(10,16)(12,14)(18,22)(23,30)(24,27)(31,32)(33,36,40,37)(34,41,38,46)(35,44,42,45)(39,48,43,47);(1,2)(3,7)(4,11)(5,8)(6,9)(10,14)(12,16)(13,15)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,37,40,36)(34,41,38,46)(35,45,42,44)(39,48,43,47);(1,4,6,7)(2,3,9,11)(5,12,13,14)(8,10,15,16)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,38,40,34)(35,43,42,39)(36,46,37,41)(44,47,45,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,39,40,43)(34,42,38,35)(36,47,37,48)(41,44,46,45)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833314);