Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232528 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2240 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)(3,5)(4,6)(7,8)(9,10)(11,21)(12,17)(13,20)(14,19)(15,18)(16,23)(22,37)(24,32)(25,31)(26,40)(27,35)(28,36)(29,34)(30,39)(33,38);(5,8)(9,11)(10,16)(12,26)(13,25)(14,23)(15,24)(17,33)(18,31)(19,32)(20,21)(22,28)(27,39)(29,38)(30,34)(35,37)(36,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,27)(16,30)(18,35)(19,34)(20,36)(21,33)(23,26)(24,38)(25,39)(31,40)(32,37);(10,18)(12,27)(16,31)(19,20)(21,32)(22,38)(26,39)(28,29)Polycyclic presentation:
2^9|e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gh,d^c=dh,e^c=ei,g^c=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72235174, #72235200, #72235210, … |
| maximal normal subgroup of | #72235174, #72235200, #72235210, … |
| maximal quotient of | #72235808 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139724);