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 | |
8030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(2,4)(5,6,9,12)(7,8,10,11)(13,14)(15,19)(16,18)(17,20);(1,2,3,4)(5,7)(6,8)(9,10)(11,12);(5,8)(6,10)(7,12)(9,11)(13,14,16,18)(15,17,20,19);(5,7)(6,11)(8,12)(9,10)(13,14)(15,17)(16,18)(19,20);(5,7)(6,8)(9,10)(11,12)(13,15)(14,17)(16,20)(18,19);(5,8)(6,10)(7,12)(9,11)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72238155, #72238161, #72238200, … |
| maximal normal subgroup of | #72238155, #72238161, #72238200, … |
| maximal quotient of | #72297460, #72297574, #72297759, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7875131);