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 | |
9568 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)(5,6,10,9)(7,11,12,8);(1,2)(3,4)(5,7,10,12)(6,11,9,8)(13,14,19,20)(15,18,17,16);(6,9)(7,12)(13,15,19,17)(14,16,20,18);(5,6,10,9)(7,11,12,8)(13,16,19,18)(14,15,20,17);(5,8,10,11)(6,12,9,7)(13,17,19,15)(14,16,20,18);(5,9,10,6)(7,11,12,8)(13,18,19,16)(14,15,20,17)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=hi,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72233745, #72233761, #72234451, … |
| maximal normal subgroup of | #72233745, #72233761, #72234451, … |
| maximal quotient of | #72256755, #72256785, #72261631, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7756855);