Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499855 |
| 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 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1536 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)(11,19)(12,23)(13,26)(15,30)(17,32)(18,27)(20,37)(21,38)(22,24)(28,40)(31,33)(35,39);(1,3,4,8)(2,5,6,7)(9,10)(11,20)(12,17)(13,27)(14,16)(15,21)(18,26)(19,37)(22,35)(23,32)(24,39)(25,36)(28,31)(29,34)(30,38)(33,40);(9,11)(10,15)(12,24)(13,28)(14,22)(16,31)(17,33)(18,35)(19,23)(20,27)(21,26)(25,38)(29,40)(30,32)(34,37)(36,39);(2,6)(5,7)(10,16)(15,31)(17,32)(18,27)(20,35)(30,33)(34,36)(37,39)Polycyclic presentation:
2^9|b^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,d^b=di,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 | #72210404, #72210417, #72267431, … |
| maximal normal subgroup of | #72210404, #72210417, #72267431, … |
| maximal quotient of | #72210355, #72210373, #72266732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407237);