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×#2204 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1074 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)(5,6,9,12)(7,17,19,31)(8,16,11,14)(10,23,22,15)(13,24,27,20)(18,34,25,33)(21,30,26,32)(28,36,29,35);(1,2,4,3)(6,12)(7,18)(8,11)(13,28)(15,23)(17,33)(19,25)(20,36)(21,26)(24,35)(27,29)(31,34);(5,7)(6,13)(8,18)(9,19)(10,24)(11,25)(12,27)(14,29)(15,31)(16,28)(17,23)(20,22)(21,36)(26,35)(30,34)(32,33);(1,3,4,2)(5,8,9,11)(6,14,12,16)(7,18,19,25)(10,21,22,26)(13,29,27,28)(15,30,23,32)(17,33,31,34)(20,35,24,36)Polycyclic presentation:
2^9|a^2=i,b^2=ei,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407992);