Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499803 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| prime radical | #32890 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1579 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)(9,10,13,15)(11,19,21,32)(12,23,24,17)(14,18,26,28)(16,29,27,22)(20,34,35,33)(25,38,39,31)(30,36,37,40);(1,3,4,8)(2,5,6,7)(10,15)(12,24)(14,26)(16,27)(19,32)(20,35)(31,38)(36,40);(9,11)(10,16)(12,20)(13,21)(14,22)(15,27)(17,30)(18,19)(23,37)(24,35)(25,36)(26,29)(28,32)(31,33)(34,38)(39,40);(2,6)(5,7)(9,12,13,24)(10,17,15,23)(11,20,21,35)(14,25,26,39)(16,30,27,37)(18,31,28,38)(19,33,32,34)(22,36,29,40)Polycyclic presentation:
2^9|a^2=i,b^2=ei,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,d^b=di,f^b=fh,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72317532, #72317579, #72317624, … |
| maximal normal subgroup of | #72317532, #72317579, #72317624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407260);