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 | |
8695 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)(7,11)(8,9)(10,12)(13,14,20,24)(15,22,17,19)(16,23,18,21)(25,27,26,28);(1,2)(3,4)(13,15,20,17)(14,19,24,22)(16,26,18,25)(21,28,23,27);(6,9)(10,11)(13,16,20,18)(14,21,24,23)(15,25,17,26)(19,27,22,28);(5,6)(7,10)(8,9)(11,12)(13,17,20,15)(14,22,24,19)(16,25,18,26)(21,27,23,28);(5,7)(6,10)(8,12)(9,11)(13,18,20,16)(14,23,24,21)(15,25,17,26)(19,27,22,28);(13,19,20,22)(14,15,24,17)(16,27,18,28)(21,25,23,26)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=i,f^2=i,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 | #72256673, #72256677, #72256678, … |
| maximal normal subgroup of | #72256673, #72256677, #72256678, … |
| maximal quotient of | #72297338, #72297434, #72297606, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7840155);