Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624824 |
| 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 | |
1355 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(10,17)(11,20)(13,16)(15,22)(18,33)(19,35)(21,34)(23,36)(24,39)(25,38)(26,37)(27,40)(29,32)(30,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,21)(12,16)(13,28)(14,17)(15,19)(18,25)(20,33)(22,29)(23,26)(24,27)(30,35)(31,32)(34,38)(36,39)(37,40);(9,11,22,39,16,26,31,38,14,25,30,37,13,24,15,20)(10,18,29,40,12,27,32,33,17,21,35,36,28,23,19,34);(3,8)(5,7)(9,12)(10,13)(11,23)(14,28)(15,29)(16,17)(18,26)(19,30)(20,36)(21,24)(22,32)(25,27)(31,35)(33,37)(34,39)(38,40);(2,6)(5,7)(9,13,14,16)(10,12,17,28)(11,24,25,26)(15,30,31,22)(18,27,21,23)(19,29,32,35)(20,37,38,39)(33,36,34,40)Polycyclic presentation:
2^9|c^2=gh,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,d^c=di,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287059, #72315178, #72315916, … |
| maximal normal subgroup of | #72287059, #72315178, #72315916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532069);