Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10,14,20,15,21,31,19)(11,16,25,35,24,33,40,34)(12,26,28,17,29,37,23,32)(13,18,27,39,30,36,22,38);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,27)(13,23)(14,25)(15,24)(17,36)(18,32)(19,34)(20,35)(21,33)(22,29)(26,39)(28,30)(31,40)(37,38);(9,12,14,28,15,29,31,23)(10,17,19,26,21,32,20,37)(11,22,25,13,24,27,40,30)(16,18,34,38,33,36,35,39);(1,4,5,3)(2,6,7,8)(9,13,14,27,15,30,31,22)(10,18,20,39,21,36,19,38)(11,23,25,12,24,28,40,29)(16,32,35,26,33,17,34,37);(11,24)(13,30)(16,33)(18,36)(22,27)(25,40)(34,35)(38,39)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=fh,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ci,e^b=ei,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239527, #72239550, #72239798, … |
| maximal normal subgroup of | #72239527, #72239550, #72239798, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281049);