Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
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 | #62759 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3240 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)(3,7)(4,6)(5,8)(9,10,14,19,15,20,13,17)(11,16,24,28,21,30,23,31)(12,18,27,34,25,33,26,32)(22,29,37,38,35,39,36,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,23)(14,24)(15,21)(17,31)(18,29)(19,28)(20,30)(25,35)(26,36)(27,37)(32,40)(33,39)(34,38);(10,17,20,19)(11,21)(16,28,30,31)(18,32,33,34)(22,35)(23,24)(29,38,39,40)(36,37);(3,8)(5,7)(9,12)(10,18)(11,22)(13,26)(14,27)(15,25)(16,29)(17,32)(19,34)(20,33)(21,35)(23,36)(24,37)(28,38)(30,39)(31,40);(2,6)(5,7)(11,21)(12,25)(16,30)(18,33)(23,24)(26,27)(28,31)(32,34)Polycyclic presentation:
2^9|a^2=h,c^2=gh,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72257161, #72279536, #72286452, … |
| maximal normal subgroup of | #72257161, #72279536, #72286452, … |
| maximal quotient of | #72264624, #72264644 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300717);