Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498886 |
| 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×#3021 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2193 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)(6,11)(8,18)(9,21)(12,24)(13,28)(14,22)(15,19)(16,23)(25,30)(26,36)(27,32)(29,35);(5,6)(7,12)(8,13)(9,22)(10,14)(11,21)(15,26)(16,32)(17,27)(18,30)(19,29)(20,28)(23,24)(25,33)(31,36)(34,35);(2,4)(5,7,10,17)(6,12,14,27)(8,19,18,15)(9,23,21,16)(11,24,22,32)(13,29,30,26)(20,31,33,34)(25,36,28,35);(5,8,10,18)(6,13,14,30)(7,15,17,19)(9,20,21,33)(11,25,22,28)(12,26,27,29)(16,31,23,34)(24,35,32,36)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=di,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 | #72256646, #72256647, #72312832, … |
| maximal normal subgroup of | #72256646, #72256647, #72312832, … |
| maximal quotient of | #72316496 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406130);