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 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3850 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18,22,24)(19,23,27,32)(20,25,28,29)(21,30,31,26)(33,34,39,44)(35,45,47,40)(36,43,38,41)(37,42,46,48);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,19,22,27)(18,23,24,32)(20,21,28,31)(25,30,29,26)(33,34)(35,45)(36,41)(37,48)(38,43)(39,44)(40,47)(42,46);(3,11)(4,12)(6,16)(8,13)(18,24)(19,27)(21,31)(25,29)(33,35,39,47)(34,40,44,45)(36,46,38,37)(41,48,43,42);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,18)(19,23)(20,29)(21,26)(22,24)(25,28)(27,32)(30,31)(33,36,39,38)(34,41,44,43)(35,46,47,37)(40,48,45,42);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,20,22,28)(18,25,24,29)(19,21,27,31)(23,30,32,26)(33,37)(34,42)(35,36)(38,47)(39,46)(40,41)(43,45)(44,48);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,21)(18,26)(19,28)(20,27)(22,31)(23,25)(24,30)(29,32)(33,38,39,36)(34,43,44,41)(35,37,47,46)(40,42,45,48)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=gh,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, 8834893);