Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623604 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ #2931 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,21)(11,16,24,33)(12,20,14,18)(13,27,30,38)(17,34,36,23)(19,26,39,29)(22,37,25,31)(28,32,40,35);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,25)(13,23)(14,22)(15,24)(17,27)(18,37)(19,32)(20,31)(21,33)(26,40)(28,29)(30,34)(35,39)(36,38);(1,4,5,2)(3,7,8,6)(9,11)(10,17)(12,22)(13,28)(14,25)(15,24)(16,27)(18,35)(19,31)(20,32)(21,36)(23,29)(26,34)(30,40)(33,38)(37,39);(3,8)(6,7)(9,11)(10,16)(12,25)(13,23)(14,22)(15,24)(17,27)(18,37)(19,32)(20,31)(21,33)(26,40)(28,29)(30,34)(35,39)(36,38);(9,12,15,14)(10,18,21,20)(11,22,24,25)(13,26,30,29)(16,31,33,37)(17,35,36,32)(19,38,39,27)(23,28,34,40)Polycyclic presentation:
2^9|a^2=fi,c^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=df,e^b=ei,g^c=gh,h^c=hi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72216688, #72225004, #72225016, … |
| maximal normal subgroup of | #72216688, #72225004, #72225016, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531043);