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 | |
6768 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)(5,6,10,12)(7,15,16,11)(8,19,18,13)(9,14,17,20)(23,29)(27,31)(30,35)(34,36);(1,2)(3,4)(5,7,10,16)(6,11,12,15)(8,17,18,9)(13,20,19,14)(22,25)(27,31)(28,32)(34,36);(6,12)(7,16)(8,18)(14,20)(21,22,26,25)(23,27,29,31)(24,32,33,28)(30,36,35,34);(5,6)(7,11)(8,13)(9,14)(10,12)(15,16)(17,20)(18,19)(21,23,26,29)(22,27,25,31)(24,30,33,35)(28,34,32,36);(5,8)(6,13)(7,17)(9,16)(10,18)(11,20)(12,19)(14,15)(21,24,26,33)(22,28,25,32)(23,30,29,35)(27,34,31,36);(5,9,10,17)(6,14,12,20)(7,18,16,8)(11,19,15,13)(21,25,26,22)(23,31,29,27)(24,28,33,32)(30,34,35,36)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,d^2=hi,e^2=hi,f^2=i,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, 7896839);