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 |
| prime radical | C2×#3517 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(2,4)(5,6,10,14)(7,9,13,11)(8,16,18,12)(15,19,20,17)(21,22,27,28)(23,26,30,25)(24,32,35,29)(31,33,36,34);(1,2,3,4)(5,7)(6,11)(8,15)(9,14)(10,13)(12,19)(16,17)(18,20)(22,28)(23,30)(29,32)(31,36);(5,8,10,18)(6,12,14,16)(7,15,13,20)(9,17,11,19)(21,23,27,30)(22,25,28,26)(24,31,35,36)(29,33,32,34);(21,24,27,35)(22,29,28,32)(23,31,30,36)(25,33,26,34);(5,7)(6,9)(8,15)(10,13)(11,14)(12,17)(16,19)(18,20)(21,25,27,26)(22,23,28,30)(24,33,35,34)(29,31,32,36);(5,9)(6,13)(7,14)(8,17)(10,11)(12,20)(15,16)(18,19)(21,26,27,25)(22,30,28,23)(24,34,35,33)(29,36,32,31)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=i,e^2=i,f^2=i,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, 7564941);