Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499831 |
| 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 | C2×B(4,𝔽2) | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,10,14)(7,11,16,25)(8,12,20,29)(9,13,23,32)(15,33,34,24)(17,26,21,30)(18,27,35,36)(19,31,22,28);(1,2,4,3)(5,7)(6,11)(8,18)(9,21)(10,16)(12,27)(13,30)(14,25)(15,19)(17,23)(20,35)(22,34)(24,28)(26,32)(29,36)(31,33);(5,8)(6,12)(7,15)(9,22)(10,20)(11,24)(13,31)(14,29)(16,34)(17,35)(18,21)(19,23)(25,33)(26,36)(27,30)(28,32);(1,3,4,2)(7,16)(11,25)(15,34)(17,21)(18,35)(24,33)(26,30)(27,36)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407976);