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 | |
6855 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,4,3)(5,6,9,8)(7,11,12,10)(13,14)(15,18)(16,19)(17,20)(21,24)(22,26)(23,27)(25,28);(5,7,9,12)(6,10,8,11)(13,14,18,15)(16,19,24,21)(17,20,26,22)(23,27,25,28);(2,3)(5,6,9,8)(7,10,12,11)(14,15)(19,21)(20,22)(27,28);(1,3,4,2)(13,15,18,14)(16,21,24,19)(17,22,26,20)(23,28,25,27);(5,6,9,8)(7,10,12,11)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27);(5,8,9,6)(7,11,12,10)(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gi,d^2=h,e^2=g,f^2=g,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, 7604471);