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 | |
6677 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)(7,15)(8,12)(9,13)(10,14)(11,18)(16,19)(17,20)(21,22,25,26)(23,29,31,27)(24,28,32,33)(30,35,36,34);(1,2)(3,4)(22,26)(27,29)(28,33)(34,35);(5,7)(6,11)(8,16)(9,17)(10,18)(12,13)(14,15)(19,20)(21,22)(23,27)(24,28)(25,26)(29,31)(30,34)(32,33)(35,36);(5,8,10,17)(6,12,14,20)(7,16,18,9)(11,13,15,19)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35);(5,9,10,16)(6,13,14,19)(7,17,18,8)(11,12,15,20)(21,24,25,32)(22,28,26,33)(23,30,31,36)(27,34,29,35);(5,6)(7,11)(8,12)(9,19)(10,14)(13,16)(15,18)(17,20)Polycyclic presentation:
2^9|a^2=i,d^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7565337);