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 | |
6696 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)(9,17)(12,16)(14,20)(18,19)(22,23)(25,33)(27,29)(31,36);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,18)(16,17)(19,20)(21,22,26,23)(24,27,32,29)(25,28,33,30)(31,34,36,35);(6,8)(11,13)(12,16)(18,19)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(5,7,10,15)(6,11,8,13)(9,14,17,20)(12,18,16,19)(21,24)(22,27)(23,29)(25,31)(26,32)(28,34)(30,35)(33,36);(5,8,10,6)(7,13,15,11)(9,16,17,12)(14,19,20,18)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36);(5,9)(6,12)(7,14)(8,16)(10,17)(11,18)(13,19)(15,20)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=i,e^2=i,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, 7626364);