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 | |
9010 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)(5,6,10,8)(7,9,12,11)(13,14,17,19)(15,20,16,18);(5,7,10,12)(6,11,8,9)(13,15,17,16)(14,18,19,20);(2,4)(6,8)(7,12)(13,14)(15,18)(16,20)(17,19);(1,2)(3,4)(5,6,10,8)(7,11,12,9)(13,16,17,15)(14,18,19,20);(5,8,10,6)(7,9,12,11);(5,9)(6,7)(8,12)(10,11)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=g,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72234458, #72234459, #72234466, … |
| maximal normal subgroup of | #72234458, #72234459, #72234466, … |
| maximal quotient of | #72261022, #72261026, #72305659 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7578682);