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 | |
7106 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)(7,11)(8,10)(9,16)(12,17)(13,14)(15,20)(18,19)(24,32)(28,33)(31,35)(34,36);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,18)(16,17)(19,20)(22,26)(23,29)(28,33)(31,35);(6,8)(7,13)(12,16)(15,19)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(5,7,10,13)(6,11,8,14)(9,15,17,19)(12,18,16,20);(5,8,10,6)(7,14,13,11)(9,16,17,12)(15,20,19,18)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,24)(22,28)(23,31)(25,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7623170);