Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61109 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2575 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(5,6,9,12,10,7,14,13)(8,11,18,20,17,15,19,16);(1,3,4,2)(5,7)(6,10)(8,16)(9,12)(11,19)(13,14)(15,18)(17,20);(5,6)(7,10)(8,11)(9,12)(13,14)(15,17)(16,19)(18,20);(5,8,9,18,10,17,14,19)(6,11,12,20,7,15,13,16);(8,17)(11,15)(16,20)(18,19)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,d^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,f^b=fi,h^b=hi,g^c=gi,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72221514, C4×#61110, #72279734 |
| maximal normal subgroup of | C4×#61110, #72279734 |
| maximal quotient of | #72221514, #72279990, #72280057 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289208);