Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500015 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62862 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10,14,18)(11,15,21,28)(12,22)(13,17,25,30)(16,24)(19,26)(20,27,33,38)(23,35)(29,36)(31,37)(32,34)(39,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,26)(17,27)(18,28)(22,34)(23,31)(24,32)(25,33)(29,37)(30,38)(35,40)(36,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,23)(14,24)(15,26)(17,29)(18,22)(20,31)(21,32)(25,36)(27,37)(28,34)(30,35)(33,39)(38,40);(3,8)(5,7)(9,13)(10,17)(11,20)(12,23)(14,25)(15,27)(16,29)(18,30)(19,31)(21,33)(22,35)(24,36)(26,37)(28,38)(32,39)(34,40);(2,6)(5,7)(13,25)(17,30)(20,33)(23,36)(27,38)(29,35)(31,39)(37,40)Polycyclic presentation:
2^9|a^2=fi,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72303133, #72303135, #72303149, … |
| maximal normal subgroup of | #72303133, #72303135, #72303149, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407212);