Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,3,4)(5,6,9,11)(7,8,12,10)(15,19)(18,20)(21,22,27,24)(23,30,28,25)(26,29,35,32)(31,36,34,33);(5,7)(6,8)(9,12)(10,11)(14,17)(18,20)(21,23,27,28)(22,25,24,30)(26,34,35,31)(29,36,32,33);(5,7,9,12)(6,10,11,8)(13,14,16,17)(15,18,19,20)(21,24,27,22)(23,25,28,30)(26,32,35,29)(31,33,34,36);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(21,25)(22,28)(23,24)(26,33)(27,30)(29,34)(31,32)(35,36);(5,8)(6,7)(9,10)(11,12)(13,15)(14,18)(16,19)(17,20)(21,26,27,35)(22,29,24,32)(23,31,28,34)(25,33,30,36)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
| maximal subgroup of | #72266871, #72266872, #72267466, … |
| maximal normal subgroup of | #72266871, #72266872, #72267466, … |
| maximal quotient of | #72266910, #72267425, #72267470, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4848880);