Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,7,9)(8,10,12,11)(14,19)(15,20)(16,17);(1,3,4,2)(6,9)(10,11)(13,14,16,19,18,20,17,15);(6,10)(8,12)(9,11)(13,15,17,20,18,19,16,14);(14,20)(15,19)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=ei,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,d^c=dh,f^c=fhi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72279482, #72279570, #72279595 |
| maximal normal subgroup of | #72279482, #72279570, #72279595 |
| maximal quotient of | #72310157, #72310166, #72342550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109974);