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×#3548 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6672 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)(7,11)(8,16)(9,10)(12,17)(13,15)(14,19)(18,20)(22,27)(24,34)(25,32)(28,31);(1,2,3,4)(6,10)(11,13)(12,16)(18,19)(22,27)(28,31)(29,35)(30,36);(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,22,26,27)(23,31,33,28)(24,35,34,29)(25,30,32,36);(21,23,26,33)(22,28,27,31)(24,32,34,25)(29,36,35,30);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20)(21,24)(22,29)(23,32)(25,33)(26,34)(27,35)(28,36)(30,31);(5,8,9,17)(6,12,10,16)(7,14,15,20)(11,18,13,19)(21,25,26,32)(22,30,27,36)(23,24,33,34)(28,29,31,35)Polycyclic presentation:
2^9|b^2=ghi,c^2=h,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7619532);