Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480684 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,17,14,18,29,36)(11,19,22,37,20,33,27,39)(12,23,25,35,26,16,28,34)(15,30,32,24,31,38,40,21);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(9,11)(10,15)(12,24)(13,27)(14,20)(16,33)(17,32)(18,31)(19,23)(21,26)(22,29)(25,30)(28,38)(34,39)(35,37)(36,40);(2,6)(5,7)(11,20)(15,31)(19,33)(21,24)(22,27)(30,38)(32,40)(37,39)Polycyclic presentation:
2^9|a^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di,f^c=fi,g^c=ghi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72210404, #72210417, #72212257, … |
| maximal normal subgroup of | #72210404, #72210417, #72212257, … |
| maximal quotient of | #72210339, #72212235, #72266739, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403408);