Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18524 |
| 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×#1619 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,9,12)(7,11,8,10)(13,14,17,20)(15,18,16,19);(6,10)(7,8)(11,12)(14,18)(15,16)(19,20);(14,19)(15,16)(18,20);(1,3,4,2)(5,7,9,8)(6,11,12,10)(13,15,17,16)(14,18,20,19)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,e^b=eh,f^b=fi,d^c=di,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214525, C2×#209860, #72257928, … |
| maximal normal subgroup of | #72214525, C2×#209860, #72257928, … |
| maximal quotient of | #72213969, #72214124, #72214781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 117056);