Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501090 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #24406 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
985 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)(11,16)(12,17)(13,15)(14,25)(18,24)(19,30)(20,29)(21,26)(22,27)(23,36)(28,33)(31,35)(32,39)(34,37)(38,40);(1,3,4,8)(2,5,6,7)(10,15)(11,19)(14,24)(16,28)(17,27)(20,32)(21,33)(23,35)(26,30)(29,38)(34,40)(37,39);(9,11)(10,16)(12,20)(13,21)(14,19,24,33)(15,26)(17,29)(18,30,25,28)(22,34)(23,32,35,40)(27,37)(31,39,36,38);(2,6)(5,7)(9,12,13,22)(10,17,15,27)(11,20,21,34)(14,23,24,35)(16,29,26,37)(18,31,25,36)(19,32,33,40)(28,38,30,39)Polycyclic presentation:
2^9|b^2=ei,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72320146 |
| maximal normal subgroup of | #72320146 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409868);