Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499867 |
| 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 | #32921 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,17)(8,15)(9,12)(10,14)(11,25)(13,19)(16,22)(18,34)(20,31)(21,30)(23,27)(24,32)(26,33)(28,36)(29,35);(1,2,4,3)(6,12)(7,18)(11,22)(13,28)(14,15)(17,33)(19,35)(20,26)(21,24)(23,36)(27,29)(31,34);(5,7)(6,13)(8,19)(9,20)(10,23)(11,26)(12,27)(14,17)(15,31)(16,28)(18,22)(21,36)(24,35)(25,29)(30,33)(32,34);(1,3,4,2)(5,8,9,10)(6,14,12,15)(7,19,20,23)(11,21,22,24)(13,17,27,31)(16,30,25,32)(18,35,26,36)(28,33,29,34)Polycyclic presentation:
2^9|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,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, 407981);