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 | |
8618 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)(15,20)(18,19);(1,2,3,4)(5,6,9,8)(7,10,11,12)(13,14,17,16)(15,19,20,18);(6,8)(10,12)(13,15)(14,18)(16,19)(17,20);(7,11)(10,12)(15,20)(18,19);(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,18)(17,20);(5,8,9,6)(7,12,11,10)(13,16,17,14)(15,18,20,19)Polycyclic presentation:
2^9|b^2=gi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72234865, #72234866, #72234880, … |
| maximal normal subgroup of | #72234865, #72234866, #72234880, … |
| maximal quotient of | #72263516, #72297356, #72297488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7686638);