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 | |
8396 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)(7,8)(9,10)(11,12)(13,14,19,16)(15,22,21,18)(17,24,25,20)(23,26,27,28);(1,2)(3,4)(13,15)(14,18)(16,22)(17,23)(19,21)(20,26)(24,28)(25,27);(6,10)(7,11)(14,16)(15,21)(17,25)(26,28);(5,7,9,11)(6,8,10,12)(13,16,19,14)(15,22,21,18)(17,24,25,20)(23,28,27,26);(5,8)(6,11)(7,10)(9,12)(13,17,19,25)(14,20,16,24)(15,23,21,27)(18,26,22,28);(13,18,19,22)(14,21,16,15)(17,26,25,28)(20,27,24,23)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72314768, #72314806, #72314942, … |
| maximal normal subgroup of | #72314768, #72314806, #72314942, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7619885);