Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232789 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#392 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1724 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)(3,6)(4,7)(5,8)(9,10)(11,19)(12,16)(13,25)(14,18)(15,22)(17,27)(20,35)(21,33)(23,31)(24,39)(26,29)(28,40)(30,36)(32,38)(34,37);(2,6)(3,4)(7,8)(9,11)(10,15)(12,20)(13,26)(14,21)(16,30)(17,33)(18,29)(19,25)(22,27)(23,37)(24,40)(28,31)(32,34)(35,39)(36,38);(2,7)(3,4)(6,8)(9,12)(10,16)(11,20)(13,24)(14,23)(15,28)(17,32)(18,31)(19,34)(21,37)(22,36)(25,39)(26,40)(27,38)(29,30)(33,35);(11,21)(12,23)(15,29)(16,31)(19,33)(22,26)(24,38)(32,39)Polycyclic presentation:
2^9|e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279646, #72279689, #72279736, … |
| maximal normal subgroup of | #72279646, #72279689, #72279736, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 140815);