Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,19,15,18,13,27)(11,16,23,31,21,30,22,37)(12,24,25,17,26,33,28,32)(20,34,36,29,35,40,38,39);(2,5)(6,8)(9,11)(10,16)(12,20)(13,22)(14,23)(15,21)(17,29)(18,30)(19,31)(24,34)(25,36)(26,35)(27,37)(28,38)(32,39)(33,40);(9,12)(10,17)(11,20)(13,28)(14,25)(15,26)(16,29)(18,32)(19,33)(21,35)(22,38)(23,36)(24,27)(30,39)(31,40)(34,37);(1,3,4,7)(2,6,5,8)(11,21)(16,30)(20,35)(22,23)(29,39)(31,37)(34,40)(36,38);(10,18)(16,30)(17,32)(19,27)(24,33)(29,39)(31,37)(34,40)Polycyclic presentation:
2^9|a^2=h,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=di,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72226742, #72282513, #72287117, … |
| maximal normal subgroup of | #72226742, #72282513, #72287117, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278194);