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 | |
6590 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)(7,8)(9,13)(10,11)(12,16)(14,15)(17,18)(19,20)(21,22,26,31)(23,27,33,28)(24,34,35,29)(25,30,36,32);(1,2,3,4)(21,23)(22,27)(24,32)(25,34)(26,33)(28,31)(29,36)(30,35);(6,11)(7,12)(13,18)(14,19)(21,23,26,33)(22,28,31,27)(24,30,35,32)(25,29,36,34);(5,7,10,12)(6,8,11,16)(9,14,17,19)(13,15,18,20)(21,23)(22,28)(24,32)(25,29)(26,33)(27,31)(30,35)(34,36);(5,8)(6,12)(7,11)(9,15)(10,16)(13,19)(14,18)(17,20)(21,24)(22,29)(23,32)(25,28)(26,35)(27,36)(30,33)(31,34);(5,9,10,17)(6,13,11,18)(7,14,12,19)(8,15,16,20)(21,25)(22,30)(23,29)(24,27)(26,36)(28,35)(31,32)(33,34)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,d^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7894510);