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 | |
9452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,3,4)(13,14)(15,16)(17,18)(19,20);(1,2)(3,4)(5,6,10,9)(7,12,8,11);(6,9)(11,12)(14,18)(15,19);(5,7,10,8)(6,11,9,12)(13,15,17,19)(14,16,18,20);(5,8,10,7)(6,12,9,11)(13,16)(14,19)(15,18)(17,20);(5,9,10,6)(7,12,8,11)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72233519, #72233543, #72233749, … |
| maximal normal subgroup of | #72233519, #72233543, #72233749, … |
| maximal quotient of | #72261614, #72261932, #72262084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7670540);