Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| socle | C22 | C2×B(4,𝔽2) |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1947 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)(5,6,8,14,11,17,9,15)(7,18,21,12,22,29,19,30)(10,16,24,31,26,13,23,32)(20,34,35,27,25,33,36,28);(1,2,3,4)(5,7)(6,12)(8,19)(9,21)(10,25)(11,22)(13,27)(14,18)(15,29)(16,28)(17,30)(20,26)(23,36)(24,35)(31,33)(32,34);(6,13)(10,26)(12,27)(14,31)(15,32)(16,17)(18,33)(20,25)(23,24)(28,30)(29,34)(35,36);(5,8,11,9)(6,14,17,15)(7,19,22,21)(10,23,26,24)(12,18,30,29)(13,31,16,32)(20,35,25,36)(27,33,28,34)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fhi,h^a=hi,e^b=ei,f^b=fhi,g^b=gi,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 | #72287849, #72315283 |
| maximal normal subgroup of | #72287849, #72315283 |
| maximal quotient of | #72287856 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401809);