Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498890 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#3026 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2295 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)(5,6,10,14)(7,15,18,29)(8,12,19,27)(9,21,24,13)(11,22,26,17)(16,31,32,36)(20,30,23,28)(25,34,35,33);(5,7)(6,11)(8,16)(9,22)(10,18)(12,25)(13,29)(14,26)(15,21)(17,24)(19,32)(20,34)(23,33)(27,35)(28,36)(30,31);(2,4)(5,8)(6,12)(7,16)(9,23)(10,19)(11,25)(13,30)(14,27)(15,31)(17,33)(18,32)(20,24)(21,28)(22,34)(26,35)(29,36);(8,19)(12,27)(16,32)(20,23)(25,35)(28,30)(31,36)(33,34)Polycyclic presentation:
2^9|a^2=i,e^2=i,b^a=be,c^a=cf,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,h^b=hi,d^c=di,e^c=eh,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72256607, #72256635, #72256645, … |
| maximal normal subgroup of | #72256607, #72256635, #72256645, … |
| maximal quotient of | #72256816, #72256817, #72263425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406120);