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 | |
6801 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)(5,6,9,12)(7,10,8,11)(13,14,18,19)(15,17,22,20)(16,21,25,27)(23,26,28,24);(5,6)(7,10)(8,11)(9,12)(13,14)(15,20)(16,21)(17,22)(18,19)(23,24)(25,27)(26,28);(2,4)(5,7,9,8)(6,10,12,11)(14,19)(15,22)(21,27)(23,28);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,18,22)(14,20,19,17)(16,23,25,28)(21,24,27,26);(5,8,9,7)(6,11,12,10)(13,16)(14,21)(15,23)(17,26)(18,25)(19,27)(20,24)(22,28);(13,17,18,20)(14,22,19,15)(16,24,25,26)(21,23,27,28)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=g,e^2=gi,f^2=i,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, 7603807);