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 | #37639 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6688 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)(21,22,26,30)(23,31,32,27)(24,28,29,25)(33,36,35,34);(1,2,3,4)(6,10)(11,13)(12,16)(18,19)(23,32)(27,31)(33,35)(34,36);(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,23,26,32)(22,27,30,31)(24,35,29,33)(25,34,28,36);(21,24,26,29)(22,28,30,25)(23,33,32,35)(27,34,31,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20)(21,22)(23,31)(24,28)(25,29)(26,30)(27,32)(33,36)(34,35);(5,8,9,17)(6,12,10,16)(7,14,15,20)(11,18,13,19)(21,25,26,28)(22,29,30,24)(23,34,32,36)(27,33,31,35)Polycyclic presentation:
2^9|a^2=h,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, 7619541);