Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C22 | C2×B(4,𝔽2) |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1987 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)(7,11)(8,13)(9,14,22,12)(10,15)(16,25)(17,24)(18,23,21,26)(19,28,33,27)(20,29)(30,35)(31,34,32,36);(1,2,3,4)(5,7)(6,11)(8,17)(9,21)(10,16)(12,26)(13,24)(14,23)(15,25)(18,22)(19,32)(20,30)(27,36)(28,34)(29,35)(31,33);(6,12)(7,16)(9,22)(11,23)(13,27)(14,15)(17,30)(19,33)(24,34)(25,26)(28,29)(35,36);(5,8,10,20)(6,13,15,29)(7,17,16,30)(9,19,22,33)(11,24,25,35)(12,27,14,28)(18,31,21,32)(23,34,26,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fhi,h^a=hi,c^b=ci,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 | #72256626, #72256628, #72256629, … |
| maximal normal subgroup of | #72256626, #72256628, #72256629, … |
| maximal quotient of | #72263707, #72263717, #72263729, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401791);