Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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×#3151 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1580 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)(6,11)(7,12)(8,9)(13,14,15,18,16,19,20,17);(1,2,3,4)(5,6,8,11,10,12,9,7)(14,17)(15,20)(18,19);(5,7,9,12,10,11,8,6);(6,12)(7,11)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=f,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=gi,d^c=dh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72279535, #102×AGL(1,ℤ/8), #72279570 |
| maximal normal subgroup of | #72279535, #102×AGL(1,ℤ/8), #72279570 |
| maximal quotient of | #72310118, #72342528, #72342591 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112438);