Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18746 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1298 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)(5,6,8,11)(7,9,12,10)(14,15)(16,17)(19,20);(1,3,4,2)(13,14,18,20)(15,16,19,17);(6,9)(7,12)(10,11)(13,15)(14,16)(17,20)(18,19);(13,16,18,17)(14,19,20,15)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,e^b=eh,f^b=fh,e^c=eh,f^c=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72279172, #72279482, #72279582 |
| maximal normal subgroup of | #72279172, #72279482, #72279582 |
| maximal quotient of | #72279385, #72310154, #72310164, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100394);