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 | |
7717 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,20)(16,24)(17,21)(18,23)(19,25)(22,26)(27,28);(1,2,3,4)(5,6,9,8)(7,11,12,10)(13,15)(14,18)(16,22)(17,23)(19,26)(20,21)(24,27)(25,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,20)(16,19)(17,21)(18,23)(22,27)(24,25)(26,28);(5,6,9,8)(7,11,12,10)(13,16,17,25)(14,19,21,24)(15,22,23,28)(18,26,20,27);(5,8,9,6)(7,11,12,10)(13,14)(15,18)(16,19)(17,21)(20,23)(22,26)(24,25)(27,28);(5,7)(6,11)(8,10)(9,12)(13,15)(14,20)(16,22)(17,23)(18,21)(19,27)(24,26)(25,28)Polycyclic presentation:
2^9|b^2=ghi,d^2=i,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72326930, #72333953 |
| maximal normal subgroup of | #72326930, #72333953 |
| maximal quotient of | #72327052, #72328346, #72329584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7859717);