Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501172 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #23903 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15)(8,12)(9,21,23,13)(10,14)(11,18,26,32)(16,29)(17,25)(19,34,22,27)(20,28)(24,31,36,33)(30,35);(1,2)(3,4)(5,7,10,17)(6,11,14,26)(8,16,20,30)(9,18,23,32)(12,24,28,36)(13,15,21,25)(19,31,22,33)(27,29,34,35);(2,4)(5,8,10,20)(6,12,14,28)(7,16,17,30)(9,22,23,19)(11,24,26,36)(13,27,21,34)(15,29,25,35)(18,33,32,31);(6,13)(7,17)(11,25)(12,27)(14,21)(15,26)(16,30)(18,32)(24,35)(28,34)(29,36)(31,33)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,f^b=fh,e^c=ehi,f^d=fh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72312231, #72312240, #72312273, … |
| maximal normal subgroup of | #72312231, #72312240, #72312273, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408374);