Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499795 |
| 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×D4 | C24 |
| prime radical | C2×#3025 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1969 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)(7,12)(8,18)(9,11)(10,14)(13,20)(15,30)(16,25)(17,27)(19,34)(21,33)(22,24)(23,32)(26,31)(28,35)(29,36);(1,2)(3,4)(6,11)(12,25)(13,18)(14,24)(23,27)(26,30)(28,34)(29,33);(2,4)(5,7,10,17)(6,12,14,27)(8,19,21,15)(9,16,22,32)(11,23,24,25)(13,28,29,26)(18,30,33,34)(20,35,36,31);(5,8,10,21)(6,13,14,29)(7,15,17,19)(9,20,22,36)(11,18,24,33)(12,26,27,28)(16,31,32,35)(23,34,25,30)Polycyclic presentation:
2^9|c^2=i,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,h^a=hi,c^b=cg,f^b=fh,d^c=di,e^c=eh,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72279120, #72279206, #72279238, … |
| maximal normal subgroup of | #72279120, #72279206, #72279238, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406997);