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 | |
7885 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(6,10)(7,12)(13,14,17,20)(15,18,23,27)(16,19,24,28)(21,25,22,26);(1,2)(3,4)(5,6,9,10)(7,8,12,11)(13,15)(14,18)(16,22)(17,23)(19,26)(20,27)(21,24)(25,28);(5,7)(6,11)(8,10)(9,12)(13,16)(14,19)(15,21)(17,24)(18,25)(20,28)(22,23)(26,27);(5,6,9,10)(7,8,12,11)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,27)(24,28);(5,7)(6,8)(9,12)(10,11)(13,16)(14,19)(15,22)(17,24)(18,26)(20,28)(21,23)(25,27);(5,8)(6,12)(7,10)(9,11)(13,15)(14,18)(16,21)(17,23)(19,25)(20,27)(22,24)(26,28)Polycyclic presentation:
2^9|a^2=gi,b^2=h,d^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 | #72330777, #72331611, #72336917 |
| maximal normal subgroup of | #72330777, #72331611, #72336917 |
| maximal quotient of | #72330826, #72330898, #72331663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7829689);