Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499849 |
| 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 | #2404 | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
930 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,17)(13,15)(14,25)(18,24)(19,31)(20,30)(21,27)(22,28)(23,36)(26,29)(32,35)(33,40)(34,38)(37,39);(1,3,5,4)(2,6,7,8)(10,15)(11,19)(14,24)(16,29)(17,28)(20,33)(21,26)(23,35)(27,31)(30,39)(34,37)(38,40);(1,4,5,3)(2,6,7,8)(9,11,13,21)(10,16,15,27)(12,20,22,34)(14,26,24,19)(17,30,28,38)(18,29,25,31)(23,37,35,33)(32,39,36,40);(2,7)(6,8)(9,12,13,22)(10,17,15,28)(11,20,21,34)(14,23,24,35)(16,30,27,38)(18,32,25,36)(19,33,26,37)(29,39,31,40)Polycyclic presentation:
2^9|b^2=ei,c^2=e,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=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72285500, #72311275, #72311286, … |
| maximal normal subgroup of | #72285500, #72311275, #72311286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407564);