Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
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×#3533 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,17,28)(8,12)(9,21,23,13)(10,14)(11,18)(16,31,32,30)(19,29,35,26)(20,27)(22,25)(24,33)(34,36);(5,7)(6,11)(8,16)(9,22)(10,17)(12,24)(13,28)(14,25)(15,21)(18,23)(19,34)(20,32)(26,30)(27,36)(29,31)(33,35);(2,4)(5,8,10,20)(6,12,14,27)(7,16,17,32)(9,19,23,35)(11,24,25,36)(13,29,21,26)(15,30,28,31)(18,33,22,34);(7,17)(11,25)(15,28)(16,32)(18,22)(24,36)(30,31)(33,34)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72233533, #72260678, #72286103, … |
| maximal normal subgroup of | #72233533, #72260678, #72286103, … |
| maximal quotient of | #72233512, #72234966 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400779);