Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624824 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
1391 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)(15,21)(18,32)(19,34)(22,35)(23,38)(24,37)(25,36)(27,31)(28,29)(30,33)(39,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,18)(12,16)(13,26)(14,17)(15,19)(20,33)(21,27)(22,25)(23,39)(24,30)(28,34)(29,31)(32,37)(35,38)(36,40);(9,11,21,38,16,25,29,37,14,24,28,36,13,23,15,20)(10,18,27,35,12,22,31,32,17,30,34,40,26,39,19,33);(3,8)(5,7)(9,12)(10,13)(11,22)(14,26)(15,27)(16,17)(18,23)(19,28)(20,35)(21,31)(24,39)(25,30)(29,34)(32,38)(33,36)(37,40);(2,6)(5,7)(9,13,14,16)(10,12,17,26)(11,23,24,25)(15,28,29,21)(18,22,30,39)(19,27,31,34)(20,36,37,38)(32,40,33,35)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,d^b=df,e^b=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287059, #72287306, #72287337, … |
| maximal normal subgroup of | #72287059, #72287306, #72287337, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532053);