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 | |
3869 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,7,11,9)(5,8,13,12)(10,14,16,15)(17,18)(19,27)(20,25)(21,24)(22,31)(23,26)(28,29)(30,32)(35,41)(39,45)(42,47)(46,48);(2,4)(7,9)(8,12)(14,15)(17,19,23,21)(18,24,26,27)(20,28,31,30)(22,29,25,32)(34,38)(39,45)(40,43)(46,48);(17,20,23,31)(18,22,26,25)(19,28,21,30)(24,29,27,32)(33,34)(35,39)(36,40)(37,38)(41,45)(42,46)(43,44)(47,48);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(17,20)(18,25)(19,28)(21,30)(22,26)(23,31)(24,32)(27,29)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47);(1,4,6,2)(3,9,11,7)(5,8,13,12)(10,14,16,15)(17,21,23,19)(18,24,26,27)(20,30,31,28)(22,29,25,32)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,22,23,25)(18,20,26,31)(19,29,21,32)(24,28,27,30)(33,36)(34,40)(35,42)(37,44)(38,43)(39,46)(41,47)(45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=h,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9817904);