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 | |
6630 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)(7,13)(8,12)(9,11)(10,15)(14,19)(16,18)(17,20)(24,32)(28,33)(30,36)(34,35);(1,2,3,4)(5,7)(6,10)(8,14)(9,15)(11,13)(12,17)(16,20)(18,19)(22,26)(27,29)(28,33)(34,35);(6,11)(10,13)(12,18)(17,19)(21,22,25,26)(23,29,31,27)(24,28,32,33)(30,35,36,34);(5,6)(7,10)(8,12)(9,11)(13,15)(14,17)(16,18)(19,20);(5,7)(6,10)(8,14)(9,15)(11,13)(12,17)(16,20)(18,19)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(5,8,9,16)(6,12,11,18)(7,14,15,20)(10,17,13,19)(21,24)(22,28)(23,30)(25,32)(26,33)(27,34)(29,35)(31,36)Polycyclic presentation:
2^9|b^2=gh,c^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618656);