Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×D4×#41 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(13,14,17,15)(16,19,20,18);(5,7,9,10)(6,8,11,12)(14,15)(18,19);(5,7,9,10)(6,8,11,12)(13,15,17,14)(16,19,20,18);(5,7,9,10)(6,8,11,12)(13,16)(14,18)(15,19)(17,20);(5,6)(7,8)(9,11)(10,12);(5,8)(6,10)(7,11)(9,12)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=g,d^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh152 conjugacy classes — view the full character table.
| maximal subgroup of | #72233750, #72233760, #72234447, … |
| maximal normal subgroup of | #72233750, #72233760, #72234447, … |
| maximal quotient of | #72261552, D4×#2339, #72263780, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534444);