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 | |
6794 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)(5,6,9,8)(7,11,12,10)(13,14,19,20)(15,21,16,18)(17,22,26,28)(23,25,24,27);(5,7,9,12)(6,10,8,11)(13,15,19,16)(14,18,20,21)(17,23,26,24)(22,27,28,25);(2,4)(5,6,9,8)(7,10,12,11)(14,20)(15,16)(22,28)(23,24);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,16,19,15)(14,21,20,18)(17,24,26,23)(22,25,28,27);(5,8,9,6)(7,11,12,10)(13,17)(14,22)(15,23)(16,24)(18,27)(19,26)(20,28)(21,25);(13,18,19,21)(14,15,20,16)(17,25,26,27)(22,24,28,23)Polycyclic presentation:
2^9|a^2=g,b^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, 7603837);