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 | |
7166 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)(7,12)(8,10)(9,17)(11,13)(14,18)(15,19)(16,20)(24,32)(28,33)(31,35)(34,36);(1,2)(3,4)(5,7,11,8)(6,12,13,10)(9,16,18,15)(14,19,17,20)(22,26)(23,29)(28,33)(31,35);(6,13)(7,8)(9,18)(19,20)(21,22)(23,30)(24,28)(25,26)(27,29)(31,36)(32,33)(34,35);(5,8,11,7)(6,10,13,12)(9,16,18,15)(14,19,17,20);(5,9)(6,14)(7,15)(8,16)(10,19)(11,18)(12,20)(13,17)(21,23)(22,27)(24,31)(25,29)(26,30)(28,34)(32,35)(33,36);(5,10)(6,7)(8,13)(9,19)(11,12)(14,15)(16,17)(18,20)(21,24)(22,28)(23,31)(25,32)(26,33)(27,34)(29,35)(30,36)Polycyclic presentation:
2^9|b^2=h,d^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, 7622989);