Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18746 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3205 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,9,11,10,7,8,12)(13,14)(15,17)(16,19)(18,20);(1,2,3,4)(5,7)(6,10)(8,12)(9,11);(5,6)(7,10)(8,11)(9,12)(14,17)(15,20)(18,19);(5,8,10,9)(6,11,7,12)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fi,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fhi,g^c=gh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72279172, #72279535, #72279546 |
| maximal normal subgroup of | #72279172, #72279535, #72279546 |
| maximal quotient of | #72279458, #72310170, #72342558, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102759);