Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#3481 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14,17,20)(15,19,16,18);(5,7,9,8)(6,10,12,11)(13,15,17,16)(14,18,20,19);(13,14)(15,18)(16,19)(17,20);(5,7,9,8)(6,11,12,10)(13,15,17,16)(14,19,20,18);(1,3,4,2)(13,16,17,15)(14,19,20,18);(5,8,9,7)(6,10,12,11)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=g,e^2=i,f^2=gh,b^a=bg,c^a=ch,e^a=eh,d^c=dh176 conjugacy classes — view the full character table.
| maximal subgroup of | #72234929, #72234933, #72237366, … |
| maximal normal subgroup of | #72234929, #72234933, #72237366, … |
| maximal quotient of | #72303236, C2×#7625635 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532831);