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 | |
7951 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)(5,6,9,12)(7,10,11,8)(13,14)(15,16)(17,25)(18,19)(20,26)(21,23)(22,28)(24,27);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(14,19)(16,23)(20,25)(24,28);(5,6)(7,8)(9,12)(10,11)(14,19)(15,21)(20,25)(22,27);(5,8)(6,11)(7,12)(9,10)(13,15)(14,16)(17,22)(18,21)(19,23)(20,24)(25,28)(26,27);(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,24)(18,23)(19,21)(20,22)(25,27)(26,28);(5,6)(7,10)(8,11)(9,12)(13,17)(14,20)(15,22)(16,24)(18,26)(19,25)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72263085, #72330777, #72331611, … |
| maximal normal subgroup of | #72263085, #72330777, #72331611, … |
| maximal quotient of | #72263099, #72263101, #72330876, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7943981);