Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499867 |
| 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×Q8 | C24 |
| prime radical | #32889 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1525 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)(11,19)(12,15)(13,25)(14,28)(16,22)(17,23)(18,24)(20,38)(21,34)(26,31)(27,35)(29,37)(30,36)(32,39)(33,40);(1,3,4,8)(2,5,6,7)(10,15)(11,20)(14,24)(16,32)(17,25)(19,37)(21,29)(22,40)(26,39)(27,30)(31,33)(34,38);(9,11)(10,16)(12,21)(13,26)(14,29)(15,31)(17,34)(18,32)(19,25)(20,24)(22,23)(27,40)(28,33)(30,39)(35,38)(36,37);(2,6)(5,7)(10,15)(16,31)(17,25)(18,28)(19,34)(32,33)(35,36)(37,38)Polycyclic presentation:
2^9|b^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72210417, #72212257, #72267418, … |
| maximal normal subgroup of | #72210417, #72212257, #72267418, … |
| maximal quotient of | #72210355, #72212212, #72266744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407221);