Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499855 |
| 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 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1890 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)(9,24)(10,13)(11,17)(12,20)(14,23)(15,27)(19,29)(21,34)(22,33)(25,31)(26,32)(28,35)(30,36);(1,2)(3,4)(5,7,11,22)(6,12,17,31)(8,19,10,21)(9,25,27,20)(13,30,16,28)(14,29,32,34)(15,33,24,18)(23,36,26,35);(2,4)(6,13)(8,10)(12,28)(14,24)(15,32)(16,17)(18,34)(19,21)(23,26)(29,33)(30,31)(35,36);(5,8)(6,14)(7,19)(9,26)(10,11)(12,29)(13,15)(16,24)(17,32)(18,28)(20,36)(21,22)(23,27)(25,35)(30,33)(31,34)Polycyclic presentation:
2^9|b^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,e^b=ei,f^b=fh,d^c=di,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 | #72212257, #72266731, #72266742, … |
| maximal normal subgroup of | #72212257, #72266731, #72266742, … |
| maximal quotient of | #72212212, #72266732, #72266754, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407058);