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 | |
7115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,4,3)(6,9)(10,12)(13,14,19,22)(15,16,25,20)(17,18,21,27)(23,26,28,24);(5,6,8,9)(7,10,11,12)(13,15,19,25)(14,20,22,16)(17,23,21,28)(18,24,27,26);(2,3)(13,16,19,20)(14,15,22,25)(17,26,21,24)(18,23,27,28);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,26)(18,23)(19,20)(21,24)(22,25)(27,28);(13,17)(14,18)(15,23)(16,26)(19,21)(20,24)(22,27)(25,28);(13,18)(14,21)(15,24)(16,23)(17,22)(19,27)(20,28)(25,26)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=i,d^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7604484);