Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511802 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
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 | |
2410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,16)(8,20)(9,12)(10,15)(11,18)(13,23)(14,25)(17,31)(19,33)(21,28)(22,30)(24,34)(26,32)(27,35)(29,36);(5,7)(6,11)(8,21)(9,24)(10,17)(12,27)(13,31)(14,29)(15,28)(16,20)(18,23)(19,25)(22,35)(26,36)(30,33)(32,34);(6,12)(9,25)(11,27)(13,32)(14,15)(16,33)(19,24)(20,30)(22,26)(28,29)(31,34)(35,36);(2,4)(7,17)(11,28)(16,31)(18,21)(19,24)(27,29)(33,34)(35,36)Polycyclic presentation:
2^9|e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,h^a=hi,d^b=di,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72261190 |
| maximal normal subgroup of | #72261190 |
| maximal quotient of | #72261939 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 418998);