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 | |
7230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(6,8)(10,11)(13,14)(15,23)(16,19)(17,26)(18,22)(20,24)(21,28)(25,27);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,15)(14,20)(16,23)(17,25)(18,28)(19,24)(21,27)(22,26);(5,6,9,8)(7,11,12,10)(14,16)(15,24)(17,28)(22,27);(5,7,9,12)(6,10,8,11)(13,16,19,14)(15,23,24,20)(17,26,28,21)(18,27,25,22);(5,8,9,6)(7,11,12,10)(13,17,19,28)(14,21,16,26)(15,25,24,18)(20,27,23,22);(13,18,19,25)(14,22,16,27)(15,17,24,28)(20,21,23,26)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=hi,d^2=h,e^2=h,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, 7893553);