Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186835 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3466 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)(3,7,8,5)(4,6)(9,10,14,20,15,18,13,19)(11,21,25,16,24,31,23,30)(12,17,26,33,27,32,28,34)(22,36,37,40,38,35,39,29);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,25)(14,23)(15,24)(17,29)(18,30)(19,31)(20,21)(26,39)(27,38)(28,37)(32,40)(33,35)(34,36);(9,12)(10,17)(11,22)(13,28)(14,26)(15,27)(16,29)(18,32)(19,34)(20,33)(21,35)(23,39)(24,38)(25,37)(30,40)(31,36);(10,18)(16,30)(17,32)(19,20)(21,31)(29,40)(33,34)(35,36)Polycyclic presentation:
2^9|a^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72235188, #72235200, #72255890, … |
| maximal normal subgroup of | #72235188, #72235200, #72255890, … |
| maximal quotient of | #72235925, #72258430, #72292810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94144);