Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499855 |
| 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×Q8 | C24 |
| prime radical | C2×#454 | C22 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1051 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)(5,6,11,10)(7,12,16,26)(8,18,21,31)(9,22,25,14)(13,24,27,20)(15,32,33,36)(17,30,23,29)(19,34,28,35);(1,2,4,3)(5,7)(6,12)(8,19)(9,23)(10,26)(11,16)(13,15)(14,30)(17,25)(18,34)(20,36)(21,28)(22,29)(24,32)(27,33)(31,35);(5,8)(6,13)(7,15)(9,24)(10,27)(11,21)(12,28)(14,31)(16,33)(17,35)(18,22)(19,26)(20,25)(23,34)(29,36)(30,32);(1,3,4,2)(7,16)(12,26)(15,33)(17,23)(19,28)(29,30)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72271412 |
| maximal normal subgroup of | #72271412 |
| maximal quotient of | #72271441 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407990);