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 | |
7597 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(13,14,18,20)(15,19,23,26)(16,17,24,25)(21,22,27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,23)(14,19,20,26)(16,21,24,27)(17,22,25,28);(6,9)(10,12)(14,20)(16,24)(19,26)(21,27);(7,11)(10,12)(13,16)(14,17)(15,21)(18,24)(19,22)(20,25)(23,27)(26,28);(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,21)(17,22)(18,23)(20,26)(24,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,17)(14,16)(15,22)(18,25)(19,21)(20,24)(23,28)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72233988, #72297211 |
| maximal normal subgroup of | #72233988, #72297211 |
| maximal quotient of | #72234010, #72297247, #72297532, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7900161);