Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499867 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1917 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)(7,18)(8,16)(9,12)(10,26)(11,17)(13,19)(14,22)(15,23)(20,34)(21,33)(24,31)(25,29)(27,32)(28,35)(30,36);(1,2)(3,4)(6,12)(8,10)(9,23)(13,30)(14,26)(15,17)(16,29)(18,34)(19,27)(20,24)(28,32)(31,33);(2,4)(5,7)(6,13)(8,19)(9,24)(10,27)(11,21)(12,28)(14,31)(15,30)(16,33)(17,32)(18,26)(20,23)(22,36)(25,35)(29,34);(5,8,11,10)(6,14,17,29)(7,19,21,27)(9,25,23,22)(12,16,15,26)(13,31,32,34)(18,28,33,30)(20,36,24,35)Polycyclic presentation:
2^9|d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fh,e^c=eh,f^c=fi,e^d=ei,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72340705, #72340712, #72340715 |
| maximal normal subgroup of | #72340705, #72340712, #72340715 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407067);