Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
6805 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(5,6)(7,10)(8,12)(9,13)(11,14)(15,18)(16,17)(19,20)(21,22,27,30)(23,28,33,36)(24,34,35,26)(25,29,31,32);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,13)(12,18)(16,19)(17,20)(21,23)(22,28)(24,25)(26,32)(27,33)(29,34)(30,36)(31,35);(7,14)(10,11)(15,20)(18,19)(21,24,27,35)(22,26,30,34)(23,31,33,25)(28,29,36,32);(5,6)(7,10)(8,16)(9,13)(11,14)(12,17)(15,19)(18,20)(21,25)(22,29)(23,24)(26,36)(27,31)(28,34)(30,32)(33,35);(5,7)(6,11)(8,15)(9,14)(10,13)(12,19)(16,18)(17,20)(21,24)(22,26)(23,25)(27,35)(28,32)(29,36)(30,34)(31,33);(5,8)(6,12)(7,15)(9,17)(10,18)(11,19)(13,16)(14,20)(21,26)(22,24)(23,32)(25,28)(27,34)(29,33)(30,35)(31,36)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7977021);