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 | |
6695 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)(21,22,26,31)(23,27,33,28)(24,32,35,29)(25,30,36,34);(1,2)(3,4)(5,6)(7,12)(8,13)(9,15)(10,11)(14,19)(16,18)(17,20)(21,23)(22,27)(24,30)(25,29)(26,33)(28,31)(32,36)(34,35);(6,11)(9,19)(12,20)(13,16)(21,23,26,33)(22,28,31,27)(24,30,35,34)(25,29,36,32);(5,7)(6,12)(8,14)(9,16)(10,17)(11,20)(13,19)(15,18)(21,23)(22,27)(24,34)(25,32)(26,33)(28,31)(29,36)(30,35);(5,8)(6,13)(7,15)(9,12)(10,18)(11,16)(14,17)(19,20)(21,24,26,35)(22,29,31,32)(23,30,33,34)(25,28,36,27);(5,9)(6,14)(7,16)(8,12)(10,19)(11,15)(13,17)(18,20)(21,25)(22,30)(23,32)(24,28)(26,36)(27,35)(29,33)(31,34)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7720313);