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 | |
6712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(5,6,10,12)(7,14,8,11)(9,17,18,13)(15,19,16,20)(24,32)(28,33)(31,35)(34,36);(1,2)(3,4)(5,7,10,8)(6,11,12,14)(9,15,18,16)(13,19,17,20)(22,26)(23,29)(28,33)(31,35);(6,12)(7,8)(13,17)(15,16)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(5,8,10,7)(6,11,12,14)(9,16,18,15)(13,19,17,20);(5,6)(7,14)(8,11)(9,13)(10,12)(15,20)(16,19)(17,18)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(5,9,10,18)(6,13,12,17)(7,15,8,16)(11,19,14,20)(21,24,25,32)(22,28,26,33)(23,31,29,35)(27,34,30,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=h,f^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, 7627113);