Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,12)(8,20)(9,14)(10,15,25,30)(11,16)(13,21,31,23)(17,27)(18,28,24,35)(19,29)(22,32)(26,33,36,34);(1,2,3,4)(5,7)(6,12)(8,17)(9,22)(10,24)(11,19)(13,26)(14,32)(15,28)(16,29)(18,25)(20,27)(21,34)(23,33)(30,35)(31,36);(5,8,11,9)(6,13,16,31)(7,17,19,22)(10,23,25,21)(12,26,29,36)(14,30,20,15)(18,34,24,33)(27,28,32,35);(5,9,11,8)(6,14,16,20)(7,17,19,22)(10,23,25,21)(12,27,29,32)(13,30,31,15)(18,33,24,34)(26,28,36,35)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fhi,h^a=hi,d^b=di,f^b=fhi,e^c=eh,f^c=fi,g^c=gh,f^e=fi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256625, #72256626, #72256630, … |
| maximal normal subgroup of | #72256625, #72256626, #72256630, … |
| maximal quotient of | #72263703, #72263709, #72263734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401795);